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

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. fma-def 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-def 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. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 4: 85.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4200000000000 \lor \neg \left(z \leq 4.3 \cdot 10^{-32}\right):\\ \;\;\;\;x + z \cdot \left(0.3333333333333333 \cdot \left(\sin y \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4200000000000.0) (not (<= z 4.3e-32)))
   (+ x (* z (* 0.3333333333333333 (* (sin y) 3.0))))
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4200000000000.0) || !(z <= 4.3e-32)) {
		tmp = x + (z * (0.3333333333333333 * (sin(y) * 3.0)));
	} 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 <= (-4200000000000.0d0)) .or. (.not. (z <= 4.3d-32))) then
        tmp = x + (z * (0.3333333333333333d0 * (sin(y) * 3.0d0)))
    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 <= -4200000000000.0) || !(z <= 4.3e-32)) {
		tmp = x + (z * (0.3333333333333333 * (Math.sin(y) * 3.0)));
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4200000000000.0) or not (z <= 4.3e-32):
		tmp = x + (z * (0.3333333333333333 * (math.sin(y) * 3.0)))
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4200000000000.0) || !(z <= 4.3e-32))
		tmp = Float64(x + Float64(z * Float64(0.3333333333333333 * Float64(sin(y) * 3.0))));
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4200000000000.0) || ~((z <= 4.3e-32)))
		tmp = x + (z * (0.3333333333333333 * (sin(y) * 3.0)));
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4200000000000.0], N[Not[LessEqual[z, 4.3e-32]], $MachinePrecision]], N[(x + N[(z * N[(0.3333333333333333 * N[(N[Sin[y], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2e12 or 4.2999999999999999e-32 < z

   1. Initial program 99.8%

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

      1. add-cbrt-cube 90.0%

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

      2. pow1/3 57.0%

         \[\leadsto x \cdot \cos y + z \cdot \color{blue}{{\left(\left(\sin y \cdot \sin y\right) \cdot \sin y\right)}^{0.3333333333333333}} \]

      3. pow3 57.0%

         \[\leadsto x \cdot \cos y + z \cdot {\color{blue}{\left({\sin y}^{3}\right)}}^{0.3333333333333333} \]

   4. Applied egg-rr 57.0%

      \[\leadsto x \cdot \cos y + z \cdot \color{blue}{{\left({\sin y}^{3}\right)}^{0.3333333333333333}} \]

   5. Taylor expanded in y around 0 52.9%

      \[\leadsto \color{blue}{x} + z \cdot {\left({\sin y}^{3}\right)}^{0.3333333333333333} \]
   6. Step-by-step derivation

      1. unpow1/3 80.7%

         \[\leadsto x + z \cdot \color{blue}{\sqrt[3]{{\sin y}^{3}}} \]

      2. rem-cbrt-cube 90.5%

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

      3. add-log-exp 70.7%

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

      4. rem-log-exp 70.7%

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

      5. add-cbrt-cube 70.6%

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

      6. pow1/3 70.7%

         \[\leadsto x + z \cdot \log \color{blue}{\left({\left(\left(\log \left(e^{e^{\sin y}}\right) \cdot \log \left(e^{e^{\sin y}}\right)\right) \cdot \log \left(e^{e^{\sin y}}\right)\right)}^{0.3333333333333333}\right)} \]

      7. log-pow 70.7%

         \[\leadsto x + z \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(\left(\log \left(e^{e^{\sin y}}\right) \cdot \log \left(e^{e^{\sin y}}\right)\right) \cdot \log \left(e^{e^{\sin y}}\right)\right)\right)} \]

      8. pow3 70.6%

         \[\leadsto x + z \cdot \left(0.3333333333333333 \cdot \log \color{blue}{\left({\log \left(e^{e^{\sin y}}\right)}^{3}\right)}\right) \]

      9. log-pow 70.7%

         \[\leadsto x + z \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \log \left(e^{e^{\sin y}}\right)\right)}\right) \]

      10. rem-log-exp 70.7%

          \[\leadsto x + z \cdot \left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(e^{\sin y}\right)}\right)\right) \]

      11. add-log-exp 90.3%

          \[\leadsto x + z \cdot \left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\sin y}\right)\right) \]

   7. Applied egg-rr 90.3%

      \[\leadsto x + z \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(3 \cdot \sin y\right)\right)} \]

   if -4.2e12 < z < 4.2999999999999999e-32

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 88.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4200000000000 \lor \neg \left(z \leq 4.3 \cdot 10^{-32}\right):\\ \;\;\;\;x + z \cdot \left(0.3333333333333333 \cdot \left(\sin y \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-5} \lor \neg \left(y \leq 0.0002\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8e-5) (not (<= y 0.0002))) (* x (cos y)) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8e-5) || !(y <= 0.0002)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (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 <= (-8d-5)) .or. (.not. (y <= 0.0002d0))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8e-5) || !(y <= 0.0002)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -8e-5) or not (y <= 0.0002):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -8e-5) || !(y <= 0.0002))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -8e-5) || ~((y <= 0.0002)))
		tmp = x * cos(y);
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -8e-5], N[Not[LessEqual[y, 0.0002]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000065e-5 or 2.0000000000000001e-4 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 56.4%

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

   if -8.00000000000000065e-5 < y < 2.0000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-5} \lor \neg \left(y \leq 0.0002\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.68 \cdot 10^{-49} \lor \neg \left(x \leq 1.35 \cdot 10^{-149}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.68e-49) (not (<= x 1.35e-149))) (* x (cos y)) (* (sin y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.68e-49) || !(x <= 1.35e-149)) {
		tmp = x * cos(y);
	} else {
		tmp = sin(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 <= (-1.68d-49)) .or. (.not. (x <= 1.35d-149))) then
        tmp = x * cos(y)
    else
        tmp = sin(y) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.68e-49) || !(x <= 1.35e-149)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = Math.sin(y) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.68e-49) or not (x <= 1.35e-149):
		tmp = x * math.cos(y)
	else:
		tmp = math.sin(y) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.68e-49) || !(x <= 1.35e-149))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(sin(y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.68e-49) || ~((x <= 1.35e-149)))
		tmp = x * cos(y);
	else
		tmp = sin(y) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.68e-49], N[Not[LessEqual[x, 1.35e-149]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6800000000000001e-49 or 1.35000000000000007e-149 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 85.6%

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

   if -1.6800000000000001e-49 < x < 1.35000000000000007e-149

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 72.4%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.68 \cdot 10^{-49} \lor \neg \left(x \leq 1.35 \cdot 10^{-149}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.1% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-138}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-129}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.8e-138) x (if (<= x 8.5e-129) (* y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.8e-138) {
		tmp = x;
	} else if (x <= 8.5e-129) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-9.8d-138)) then
        tmp = x
    else if (x <= 8.5d-129) then
        tmp = y * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.8e-138) {
		tmp = x;
	} else if (x <= 8.5e-129) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9.8e-138:
		tmp = x
	elif x <= 8.5e-129:
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.8e-138)
		tmp = x;
	elseif (x <= 8.5e-129)
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9.8e-138)
		tmp = x;
	elseif (x <= 8.5e-129)
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.8e-138], x, If[LessEqual[x, 8.5e-129], N[(y * z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -9.80000000000000033e-138 or 8.49999999999999937e-129 < x

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. fma-def 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 51.7%

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

   if -9.80000000000000033e-138 < x < 8.49999999999999937e-129

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0 48.4%

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

      1. +-commutative 48.4%

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

   5. Simplified 48.4%

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

   6. Taylor expanded in y around inf 34.6%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-138}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-129}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.8% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0 54.0%

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

   1. +-commutative 54.0%

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

5. Simplified 54.0%

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

6. Final simplification 54.0%

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

Alternative 9: 39.7% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. fma-def 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in y around 0 42.1%

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

6. Final simplification 42.1%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))