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

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 10

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \sin y, z \cdot \cos y\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 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 99.8%

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

3. Final simplification 99.8%

   \[\leadsto z \cdot \cos y + x \cdot \sin y \]
4. Add Preprocessing

Alternative 3: 73.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \left(y + z \cdot \frac{\cos y}{x}\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-10} \lor \neg \left(x \leq 3.8 \cdot 10^{+46}\right) \land x \leq 1.05 \cdot 10^{+115}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))))
   (if (<= x -1.65e+189)
     t_0
     (if (<= x -9.5e-53)
       (* x (+ y (* z (/ (cos y) x))))
       (if (or (<= x 1.8e-10) (and (not (<= x 3.8e+46)) (<= x 1.05e+115)))
         (* z (cos y))
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double tmp;
	if (x <= -1.65e+189) {
		tmp = t_0;
	} else if (x <= -9.5e-53) {
		tmp = x * (y + (z * (cos(y) / x)));
	} else if ((x <= 1.8e-10) || (!(x <= 3.8e+46) && (x <= 1.05e+115))) {
		tmp = z * cos(y);
	} 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) :: tmp
    t_0 = x * sin(y)
    if (x <= (-1.65d+189)) then
        tmp = t_0
    else if (x <= (-9.5d-53)) then
        tmp = x * (y + (z * (cos(y) / x)))
    else if ((x <= 1.8d-10) .or. (.not. (x <= 3.8d+46)) .and. (x <= 1.05d+115)) then
        tmp = z * cos(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.sin(y);
	double tmp;
	if (x <= -1.65e+189) {
		tmp = t_0;
	} else if (x <= -9.5e-53) {
		tmp = x * (y + (z * (Math.cos(y) / x)));
	} else if ((x <= 1.8e-10) || (!(x <= 3.8e+46) && (x <= 1.05e+115))) {
		tmp = z * Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.sin(y)
	tmp = 0
	if x <= -1.65e+189:
		tmp = t_0
	elif x <= -9.5e-53:
		tmp = x * (y + (z * (math.cos(y) / x)))
	elif (x <= 1.8e-10) or (not (x <= 3.8e+46) and (x <= 1.05e+115)):
		tmp = z * math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	tmp = 0.0
	if (x <= -1.65e+189)
		tmp = t_0;
	elseif (x <= -9.5e-53)
		tmp = Float64(x * Float64(y + Float64(z * Float64(cos(y) / x))));
	elseif ((x <= 1.8e-10) || (!(x <= 3.8e+46) && (x <= 1.05e+115)))
		tmp = Float64(z * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * sin(y);
	tmp = 0.0;
	if (x <= -1.65e+189)
		tmp = t_0;
	elseif (x <= -9.5e-53)
		tmp = x * (y + (z * (cos(y) / x)));
	elseif ((x <= 1.8e-10) || (~((x <= 3.8e+46)) && (x <= 1.05e+115)))
		tmp = z * cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.65e+189], t$95$0, If[LessEqual[x, -9.5e-53], N[(x * N[(y + N[(z * N[(N[Cos[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 1.8e-10], And[N[Not[LessEqual[x, 3.8e+46]], $MachinePrecision], LessEqual[x, 1.05e+115]]], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6500000000000001e189 or 1.8e-10 < x < 3.7999999999999999e46 or 1.05000000000000002e115 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 79.5%

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

   if -1.6500000000000001e189 < x < -9.5000000000000008e-53

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. add-cube-cbrt 98.6%

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

      4. associate-*r* 98.6%

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

      5. fma-define 98.6%

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

      6. pow2 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in y around 0 68.5%

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

   6. Taylor expanded in x around inf 69.2%

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

      1. associate-/l* 69.1%

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

   8. Simplified 69.1%

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

   if -9.5000000000000008e-53 < x < 1.8e-10 or 3.7999999999999999e46 < x < 1.05000000000000002e115

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 90.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+189}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \left(y + z \cdot \frac{\cos y}{x}\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-10} \lor \neg \left(x \leq 3.8 \cdot 10^{+46}\right) \land x \leq 1.05 \cdot 10^{+115}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := z \cdot \left(1 + x \cdot \frac{\sin y}{z}\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))) (t_1 (* z (+ 1.0 (* x (/ (sin y) z))))))
   (if (<= x -5e-52)
     t_1
     (if (<= x 8.8e-13)
       t_0
       (if (<= x 2.1e+42) t_1 (if (<= x 6.6e+113) t_0 (* x (sin y))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double t_1 = z * (1.0 + (x * (sin(y) / z)));
	double tmp;
	if (x <= -5e-52) {
		tmp = t_1;
	} else if (x <= 8.8e-13) {
		tmp = t_0;
	} else if (x <= 2.1e+42) {
		tmp = t_1;
	} else if (x <= 6.6e+113) {
		tmp = t_0;
	} else {
		tmp = x * sin(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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * cos(y)
    t_1 = z * (1.0d0 + (x * (sin(y) / z)))
    if (x <= (-5d-52)) then
        tmp = t_1
    else if (x <= 8.8d-13) then
        tmp = t_0
    else if (x <= 2.1d+42) then
        tmp = t_1
    else if (x <= 6.6d+113) then
        tmp = t_0
    else
        tmp = x * sin(y)
    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 = z * (1.0 + (x * (Math.sin(y) / z)));
	double tmp;
	if (x <= -5e-52) {
		tmp = t_1;
	} else if (x <= 8.8e-13) {
		tmp = t_0;
	} else if (x <= 2.1e+42) {
		tmp = t_1;
	} else if (x <= 6.6e+113) {
		tmp = t_0;
	} else {
		tmp = x * Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	t_1 = z * (1.0 + (x * (math.sin(y) / z)))
	tmp = 0
	if x <= -5e-52:
		tmp = t_1
	elif x <= 8.8e-13:
		tmp = t_0
	elif x <= 2.1e+42:
		tmp = t_1
	elif x <= 6.6e+113:
		tmp = t_0
	else:
		tmp = x * math.sin(y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	t_1 = Float64(z * Float64(1.0 + Float64(x * Float64(sin(y) / z))))
	tmp = 0.0
	if (x <= -5e-52)
		tmp = t_1;
	elseif (x <= 8.8e-13)
		tmp = t_0;
	elseif (x <= 2.1e+42)
		tmp = t_1;
	elseif (x <= 6.6e+113)
		tmp = t_0;
	else
		tmp = Float64(x * sin(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	t_1 = z * (1.0 + (x * (sin(y) / z)));
	tmp = 0.0;
	if (x <= -5e-52)
		tmp = t_1;
	elseif (x <= 8.8e-13)
		tmp = t_0;
	elseif (x <= 2.1e+42)
		tmp = t_1;
	elseif (x <= 6.6e+113)
		tmp = t_0;
	else
		tmp = x * sin(y);
	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[(z * N[(1.0 + N[(x * N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e-52], t$95$1, If[LessEqual[x, 8.8e-13], t$95$0, If[LessEqual[x, 2.1e+42], t$95$1, If[LessEqual[x, 6.6e+113], t$95$0, N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5e-52 or 8.79999999999999986e-13 < x < 2.09999999999999995e42

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf 94.3%

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

      1. associate-/l* 94.3%

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

   5. Simplified 94.3%

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

      1. distribute-rgt-in 94.3%

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

      2. +-commutative 94.3%

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

      3. *-commutative 94.3%

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

      4. associate-*l* 84.3%

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

      5. fma-define 84.3%

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

      6. *-commutative 84.3%

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

   7. Applied egg-rr 84.3%

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

   8. Taylor expanded in y around 0 73.2%

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

   9. Taylor expanded in z around inf 77.8%

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

       1. associate-*r/ 77.8%

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

   11. Simplified 77.8%

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

   if -5e-52 < x < 8.79999999999999986e-13 or 2.09999999999999995e42 < x < 6.6000000000000006e113

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 90.6%

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

   if 6.6000000000000006e113 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 79.1%

      \[\leadsto \color{blue}{x \cdot \sin y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+26} \lor \neg \left(z \leq 62000000\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z + \frac{\sin y}{z} \cdot \left(x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.5e+26) (not (<= z 62000000.0)))
   (* z (cos y))
   (+ z (* (/ (sin y) z) (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.5e+26) || !(z <= 62000000.0)) {
		tmp = z * cos(y);
	} else {
		tmp = z + ((sin(y) / z) * (x * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.5d+26)) .or. (.not. (z <= 62000000.0d0))) then
        tmp = z * cos(y)
    else
        tmp = z + ((sin(y) / z) * (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.5e+26) || !(z <= 62000000.0)) {
		tmp = z * Math.cos(y);
	} else {
		tmp = z + ((Math.sin(y) / z) * (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.5e+26) or not (z <= 62000000.0):
		tmp = z * math.cos(y)
	else:
		tmp = z + ((math.sin(y) / z) * (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.5e+26) || !(z <= 62000000.0))
		tmp = Float64(z * cos(y));
	else
		tmp = Float64(z + Float64(Float64(sin(y) / z) * Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.5e+26) || ~((z <= 62000000.0)))
		tmp = z * cos(y);
	else
		tmp = z + ((sin(y) / z) * (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.5e+26], N[Not[LessEqual[z, 62000000.0]], $MachinePrecision]], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(z + N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.49999999999999999e26 or 6.2e7 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 86.5%

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

   if -1.49999999999999999e26 < z < 6.2e7

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf 91.3%

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

      1. associate-/l* 91.3%

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

   5. Simplified 91.3%

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

      1. distribute-rgt-in 91.3%

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

      2. +-commutative 91.3%

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

      3. *-commutative 91.3%

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

      4. associate-*l* 95.8%

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

      5. fma-define 95.8%

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

      6. *-commutative 95.8%

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

   7. Applied egg-rr 95.8%

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

   8. Taylor expanded in y around 0 81.4%

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

      1. fma-undefine 81.4%

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

      2. *-commutative 81.4%

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

   10. Applied egg-rr 81.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+26} \lor \neg \left(z \leq 62000000\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z + \frac{\sin y}{z} \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.000425 \lor \neg \left(y \leq 68000000000\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5 + -0.16666666666666666 \cdot \left(x \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.000425) (not (<= y 68000000000.0)))
   (* z (cos y))
   (+ z (* y (+ x (* y (+ (* z -0.5) (* -0.16666666666666666 (* x y)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.000425) || !(y <= 68000000000.0)) {
		tmp = z * cos(y);
	} else {
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (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 <= (-0.000425d0)) .or. (.not. (y <= 68000000000.0d0))) then
        tmp = z * cos(y)
    else
        tmp = z + (y * (x + (y * ((z * (-0.5d0)) + ((-0.16666666666666666d0) * (x * y))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.000425) || !(y <= 68000000000.0)) {
		tmp = z * Math.cos(y);
	} else {
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.000425) or not (y <= 68000000000.0):
		tmp = z * math.cos(y)
	else:
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.000425) || !(y <= 68000000000.0))
		tmp = Float64(z * cos(y));
	else
		tmp = Float64(z + Float64(y * Float64(x + Float64(y * Float64(Float64(z * -0.5) + Float64(-0.16666666666666666 * Float64(x * y)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.000425) || ~((y <= 68000000000.0)))
		tmp = z * cos(y);
	else
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.000425], N[Not[LessEqual[y, 68000000000.0]], $MachinePrecision]], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(z + N[(y * N[(x + N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(-0.16666666666666666 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.24999999999999976e-4 or 6.8e10 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 61.9%

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

   if -4.24999999999999976e-4 < y < 6.8e10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.000425 \lor \neg \left(y \leq 68000000000\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5 + -0.16666666666666666 \cdot \left(x \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.32 \lor \neg \left(y \leq 0.1\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5 + -0.16666666666666666 \cdot \left(x \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.32) (not (<= y 0.1)))
   (* x (sin y))
   (+ z (* y (+ x (* y (+ (* z -0.5) (* -0.16666666666666666 (* x y)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.32) || !(y <= 0.1)) {
		tmp = x * sin(y);
	} else {
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (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 <= (-0.32d0)) .or. (.not. (y <= 0.1d0))) then
        tmp = x * sin(y)
    else
        tmp = z + (y * (x + (y * ((z * (-0.5d0)) + ((-0.16666666666666666d0) * (x * y))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.32) || !(y <= 0.1)) {
		tmp = x * Math.sin(y);
	} else {
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.32) or not (y <= 0.1):
		tmp = x * math.sin(y)
	else:
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.32) || !(y <= 0.1))
		tmp = Float64(x * sin(y));
	else
		tmp = Float64(z + Float64(y * Float64(x + Float64(y * Float64(Float64(z * -0.5) + Float64(-0.16666666666666666 * Float64(x * y)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.32) || ~((y <= 0.1)))
		tmp = x * sin(y);
	else
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.32], N[Not[LessEqual[y, 0.1]], $MachinePrecision]], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(z + N[(y * N[(x + N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(-0.16666666666666666 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.320000000000000007 or 0.10000000000000001 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 41.0%

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

   if -0.320000000000000007 < y < 0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.4%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.32 \lor \neg \left(y \leq 0.1\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5 + -0.16666666666666666 \cdot \left(x \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.9% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+220} \lor \neg \left(x \leq 5.2 \cdot 10^{+135}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.8e+220) (not (<= x 5.2e+135))) (* x y) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.8e+220) || !(x <= 5.2e+135)) {
		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 ((x <= (-8.8d+220)) .or. (.not. (x <= 5.2d+135))) 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 ((x <= -8.8e+220) || !(x <= 5.2e+135)) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.8e+220) or not (x <= 5.2e+135):
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.8e+220) || !(x <= 5.2e+135))
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.8e+220) || ~((x <= 5.2e+135)))
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.8e+220], N[Not[LessEqual[x, 5.2e+135]], $MachinePrecision]], N[(x * y), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -8.79999999999999957e220 or 5.2e135 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 83.6%

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

   4. Taylor expanded in y around 0 41.1%

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

   if -8.79999999999999957e220 < x < 5.2e135

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. add-cube-cbrt 98.2%

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

      4. associate-*r* 98.2%

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

      5. fma-define 98.2%

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

      6. pow2 98.2%

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in y around 0 40.6%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+220} \lor \neg \left(x \leq 5.2 \cdot 10^{+135}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.3% 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 99.8%

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

3. Taylor expanded in y around 0 47.3%

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

   1. +-commutative 47.3%

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

5. Simplified 47.3%

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

6. Final simplification 47.3%

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

Alternative 10: 39.5% 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 99.8%

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

   1. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. add-cube-cbrt 98.4%

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

   4. associate-*r* 98.4%

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

   5. fma-define 98.4%

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

   6. pow2 98.4%

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in y around 0 34.8%

   \[\leadsto \color{blue}{z} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024091 
(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))))