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

Percentage Accurate: 99.8% → 99.8%

Time: 7.2s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := x \cdot \sin y\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+243}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.25:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 30500:\\ \;\;\;\;-0.5 \cdot \left(y \cdot \left(y \cdot z\right)\right) + \left(z + x \cdot y\right)\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+209} \lor \neg \left(y \leq 1.8 \cdot 10^{+233}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))) (t_1 (* x (sin y))))
   (if (<= y -2.6e+243)
     t_0
     (if (<= y -0.25)
       t_1
       (if (<= y 30500.0)
         (+ (* -0.5 (* y (* y z))) (+ z (* x y)))
         (if (or (<= y 3e+209) (not (<= y 1.8e+233))) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double t_1 = x * sin(y);
	double tmp;
	if (y <= -2.6e+243) {
		tmp = t_0;
	} else if (y <= -0.25) {
		tmp = t_1;
	} else if (y <= 30500.0) {
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y));
	} else if ((y <= 3e+209) || !(y <= 1.8e+233)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * cos(y)
    t_1 = x * sin(y)
    if (y <= (-2.6d+243)) then
        tmp = t_0
    else if (y <= (-0.25d0)) then
        tmp = t_1
    else if (y <= 30500.0d0) then
        tmp = ((-0.5d0) * (y * (y * z))) + (z + (x * y))
    else if ((y <= 3d+209) .or. (.not. (y <= 1.8d+233))) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.cos(y);
	double t_1 = x * Math.sin(y);
	double tmp;
	if (y <= -2.6e+243) {
		tmp = t_0;
	} else if (y <= -0.25) {
		tmp = t_1;
	} else if (y <= 30500.0) {
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y));
	} else if ((y <= 3e+209) || !(y <= 1.8e+233)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	t_1 = x * math.sin(y)
	tmp = 0
	if y <= -2.6e+243:
		tmp = t_0
	elif y <= -0.25:
		tmp = t_1
	elif y <= 30500.0:
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y))
	elif (y <= 3e+209) or not (y <= 1.8e+233):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	t_1 = Float64(x * sin(y))
	tmp = 0.0
	if (y <= -2.6e+243)
		tmp = t_0;
	elseif (y <= -0.25)
		tmp = t_1;
	elseif (y <= 30500.0)
		tmp = Float64(Float64(-0.5 * Float64(y * Float64(y * z))) + Float64(z + Float64(x * y)));
	elseif ((y <= 3e+209) || !(y <= 1.8e+233))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	t_1 = x * sin(y);
	tmp = 0.0;
	if (y <= -2.6e+243)
		tmp = t_0;
	elseif (y <= -0.25)
		tmp = t_1;
	elseif (y <= 30500.0)
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y));
	elseif ((y <= 3e+209) || ~((y <= 1.8e+233)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.6e+243], t$95$0, If[LessEqual[y, -0.25], t$95$1, If[LessEqual[y, 30500.0], N[(N[(-0.5 * N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 3e+209], N[Not[LessEqual[y, 1.8e+233]], $MachinePrecision]], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.59999999999999997e243 or 2.99999999999999985e209 < y < 1.7999999999999999e233

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. add-cube-cbrt 99.2%

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

      3. associate-*l* 99.4%

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

      4. fma-def 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right)} \]

      5. pow2 99.4%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right) \]

   3. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right)} \]

   4. Taylor expanded in x around 0 75.1%

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

   if -2.59999999999999997e243 < y < -0.25 or 30500 < y < 2.99999999999999985e209 or 1.7999999999999999e233 < y

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. add-cube-cbrt 98.6%

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

      3. associate-*l* 98.6%

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

      4. fma-def 98.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right)} \]

      5. pow2 98.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right) \]

   3. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right)} \]

   4. Taylor expanded in x around inf 64.0%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\sin y \cdot x\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 64.0%

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

      2. *-lft-identity 64.0%

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

   6. Simplified 64.0%

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

   if -0.25 < y < 30500

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.0%

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

      1. expm1-log1p-u 90.4%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2} \cdot z\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-udef 90.1%

         \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({y}^{2} \cdot z\right)} - 1\right)} + \left(y \cdot x + z\right) \]

      3. unpow2 90.1%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right)} - 1\right) + \left(y \cdot x + z\right) \]

      4. associate-*l* 90.1%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left(y \cdot z\right)}\right)} - 1\right) + \left(y \cdot x + z\right) \]

   4. Applied egg-rr 90.1%

      \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left(y \cdot z\right)\right)} - 1\right)} + \left(y \cdot x + z\right) \]
   5. Step-by-step derivation

      1. expm1-def 90.4%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(y \cdot z\right)\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-log1p 98.0%

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

   6. Simplified 98.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+243}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq -0.25:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq 30500:\\ \;\;\;\;-0.5 \cdot \left(y \cdot \left(y \cdot z\right)\right) + \left(z + x \cdot y\right)\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+209} \lor \neg \left(y \leq 1.8 \cdot 10^{+233}\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]

Alternative 3: 85.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+39} \lor \neg \left(x \leq 10^{-19}\right):\\ \;\;\;\;z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.1e+39) (not (<= x 1e-19)))
   (+ z (* x (sin y)))
   (* z (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.1e+39) || !(x <= 1e-19)) {
		tmp = z + (x * sin(y));
	} else {
		tmp = z * 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 ((x <= (-1.1d+39)) .or. (.not. (x <= 1d-19))) then
        tmp = z + (x * sin(y))
    else
        tmp = z * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.1e+39) || !(x <= 1e-19)) {
		tmp = z + (x * Math.sin(y));
	} else {
		tmp = z * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.1e+39) or not (x <= 1e-19):
		tmp = z + (x * math.sin(y))
	else:
		tmp = z * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.1e+39) || !(x <= 1e-19))
		tmp = Float64(z + Float64(x * sin(y)));
	else
		tmp = Float64(z * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.1e+39) || ~((x <= 1e-19)))
		tmp = z + (x * sin(y));
	else
		tmp = z * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.1e+39], N[Not[LessEqual[x, 1e-19]], $MachinePrecision]], N[(z + N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001e39 or 9.9999999999999998e-20 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 93.2%

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

   if -1.1000000000000001e39 < x < 9.9999999999999998e-20

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. add-cube-cbrt 99.7%

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

      3. associate-*l* 99.7%

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

      4. fma-def 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right)} \]

      5. pow2 99.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right) \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right)} \]

   4. Taylor expanded in x around 0 88.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+39} \lor \neg \left(x \leq 10^{-19}\right):\\ \;\;\;\;z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]

Alternative 4: 75.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.06 \lor \neg \left(y \leq 0.0078\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(y \cdot \left(y \cdot z\right)\right) + \left(z + x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.06) (not (<= y 0.0078)))
   (* z (cos y))
   (+ (* -0.5 (* y (* y z))) (+ z (* x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.06) || !(y <= 0.0078)) {
		tmp = z * cos(y);
	} else {
		tmp = (-0.5 * (y * (y * z))) + (z + (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 <= (-0.06d0)) .or. (.not. (y <= 0.0078d0))) then
        tmp = z * cos(y)
    else
        tmp = ((-0.5d0) * (y * (y * z))) + (z + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.06) || !(y <= 0.0078)) {
		tmp = z * Math.cos(y);
	} else {
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.06) or not (y <= 0.0078):
		tmp = z * math.cos(y)
	else:
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.06) || !(y <= 0.0078))
		tmp = Float64(z * cos(y));
	else
		tmp = Float64(Float64(-0.5 * Float64(y * Float64(y * z))) + Float64(z + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.06) || ~((y <= 0.0078)))
		tmp = z * cos(y);
	else
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.06], N[Not[LessEqual[y, 0.0078]], $MachinePrecision]], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.059999999999999998 or 0.0077999999999999996 < y

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. add-cube-cbrt 98.8%

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

      3. associate-*l* 98.8%

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

      4. fma-def 98.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right)} \]

      5. pow2 98.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right) \]

   3. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right)} \]

   4. Taylor expanded in x around 0 47.3%

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

   if -0.059999999999999998 < y < 0.0077999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.9%

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

      1. expm1-log1p-u 92.0%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2} \cdot z\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-udef 92.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({y}^{2} \cdot z\right)} - 1\right)} + \left(y \cdot x + z\right) \]

      3. unpow2 92.0%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right)} - 1\right) + \left(y \cdot x + z\right) \]

      4. associate-*l* 92.0%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left(y \cdot z\right)}\right)} - 1\right) + \left(y \cdot x + z\right) \]

   4. Applied egg-rr 92.0%

      \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left(y \cdot z\right)\right)} - 1\right)} + \left(y \cdot x + z\right) \]
   5. Step-by-step derivation

      1. expm1-def 92.0%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(y \cdot z\right)\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-log1p 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.06 \lor \neg \left(y \leq 0.0078\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(y \cdot \left(y \cdot z\right)\right) + \left(z + x \cdot y\right)\\ \end{array} \]

Alternative 5: 42.5% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-112}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-193}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.15e-112) z (if (<= z 4.3e-193) (* x y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15e-112) {
		tmp = z;
	} else if (z <= 4.3e-193) {
		tmp = x * y;
	} else {
		tmp = 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 <= (-1.15d-112)) then
        tmp = z
    else if (z <= 4.3d-193) then
        tmp = x * y
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15e-112) {
		tmp = z;
	} else if (z <= 4.3e-193) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.15e-112:
		tmp = z
	elif z <= 4.3e-193:
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.15e-112)
		tmp = z;
	elseif (z <= 4.3e-193)
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.15e-112)
		tmp = z;
	elseif (z <= 4.3e-193)
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.15e-112], z, If[LessEqual[z, 4.3e-193], N[(x * y), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -1.14999999999999995e-112 or 4.3000000000000002e-193 < z

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. add-cube-cbrt 99.3%

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

      3. associate-*l* 99.4%

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

      4. fma-def 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right)} \]

      5. pow2 99.4%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right) \]

   3. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right)} \]

   4. Taylor expanded in y around 0 52.2%

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

   if -1.14999999999999995e-112 < z < 4.3000000000000002e-193

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 49.3%

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

   3. Taylor expanded in y around inf 33.2%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-112}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-193}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 6: 52.5% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0 57.7%

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

3. Final simplification 57.7%

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

Alternative 7: 38.8% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. add-cube-cbrt 99.1%

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

   3. associate-*l* 99.1%

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

   4. fma-def 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right)} \]

   5. pow2 99.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right) \]

3. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y} \cdot x, z \cdot \cos y\right)} \]

4. Taylor expanded in y around 0 44.0%

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

5. Final simplification 44.0%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023268 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))