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

Percentage Accurate: 99.8% → 99.8%

Time: 9.2s

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, 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 99.7%

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

   1. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

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 99.7%

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

2. Final simplification 99.7%

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

Alternative 3: 74.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0001 \lor \neg \left(y \leq 42000\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0001) (not (<= y 42000.0))) (* x (sin y)) (+ z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0001) || !(y <= 42000.0)) {
		tmp = x * sin(y);
	} else {
		tmp = 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 <= (-0.0001d0)) .or. (.not. (y <= 42000.0d0))) then
        tmp = x * sin(y)
    else
        tmp = z + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0001) || !(y <= 42000.0)) {
		tmp = x * Math.sin(y);
	} else {
		tmp = z + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0001) or not (y <= 42000.0):
		tmp = x * math.sin(y)
	else:
		tmp = z + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0001) || !(y <= 42000.0))
		tmp = Float64(x * sin(y));
	else
		tmp = Float64(z + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.0001) || ~((y <= 42000.0)))
		tmp = x * sin(y);
	else
		tmp = z + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0001], N[Not[LessEqual[y, 42000.0]], $MachinePrecision]], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000005e-4 or 42000 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in x around inf 47.6%

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

   if -1.00000000000000005e-4 < y < 42000

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.0%

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

      1. *-commutative 99.0%

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

   4. Simplified 99.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0001 \lor \neg \left(y \leq 42000\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]

Alternative 4: 73.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.42 \cdot 10^{-33} \lor \neg \left(z \leq 8200000000\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.42e-33) (not (<= z 8200000000.0)))
   (* z (cos y))
   (* x (sin y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.42e-33) || !(z <= 8200000000.0)) {
		tmp = z * cos(y);
	} 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) :: tmp
    if ((z <= (-1.42d-33)) .or. (.not. (z <= 8200000000.0d0))) then
        tmp = z * cos(y)
    else
        tmp = x * sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.42e-33) || !(z <= 8200000000.0)) {
		tmp = z * Math.cos(y);
	} else {
		tmp = x * Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.42e-33) or not (z <= 8200000000.0):
		tmp = z * math.cos(y)
	else:
		tmp = x * math.sin(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.42e-33) || !(z <= 8200000000.0))
		tmp = Float64(z * cos(y));
	else
		tmp = Float64(x * sin(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.42e-33) || ~((z <= 8200000000.0)))
		tmp = z * cos(y);
	else
		tmp = x * sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.42e-33], N[Not[LessEqual[z, 8200000000.0]], $MachinePrecision]], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.42000000000000007e-33 or 8.2e9 < z

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 86.1%

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

   if -1.42000000000000007e-33 < z < 8.2e9

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 76.5%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.42 \cdot 10^{-33} \lor \neg \left(z \leq 8200000000\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \]

Alternative 5: 40.9% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-185}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-52}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.5e-185) z (if (<= z 9.8e-52) (* x y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5e-185) {
		tmp = z;
	} else if (z <= 9.8e-52) {
		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 <= (-5.5d-185)) then
        tmp = z
    else if (z <= 9.8d-52) 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 <= -5.5e-185) {
		tmp = z;
	} else if (z <= 9.8e-52) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.5e-185:
		tmp = z
	elif z <= 9.8e-52:
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.5e-185)
		tmp = z;
	elseif (z <= 9.8e-52)
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.5e-185)
		tmp = z;
	elseif (z <= 9.8e-52)
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.5e-185], z, If[LessEqual[z, 9.8e-52], N[(x * y), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4999999999999998e-185 or 9.80000000000000037e-52 < z

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 44.1%

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

      1. +-commutative 44.1%

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

      2. associate-+l+ 44.1%

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

      3. associate-*r* 44.1%

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

      4. *-commutative 44.1%

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

      5. unpow2 44.1%

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

      6. *-commutative 44.1%

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

      7. fma-def 44.1%

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

   4. Simplified 44.1%

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

   5. Taylor expanded in y around 0 40.1%

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

   if -5.4999999999999998e-185 < z < 9.80000000000000037e-52

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 46.4%

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

      1. +-commutative 46.4%

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

      2. associate-+l+ 46.4%

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

      3. associate-*r* 46.4%

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

      4. *-commutative 46.4%

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

      5. unpow2 46.4%

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

      6. *-commutative 46.4%

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

      7. fma-def 46.4%

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

   4. Simplified 46.4%

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

   5. Taylor expanded in z around 0 34.9%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-185}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-52}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 6: 52.4% 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 99.7%

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

2. Taylor expanded in y around 0 46.2%

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

   1. *-commutative 46.2%

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

4. Simplified 46.2%

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

5. Final simplification 46.2%

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

Alternative 7: 39.2% 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 99.7%

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

2. Taylor expanded in y around 0 44.8%

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

   1. +-commutative 44.8%

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

   2. associate-+l+ 44.8%

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

   3. associate-*r* 44.8%

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

   4. *-commutative 44.8%

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

   5. unpow2 44.8%

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

   6. *-commutative 44.8%

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

   7. fma-def 44.8%

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

4. Simplified 44.8%

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

5. Taylor expanded in y around 0 32.7%

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

6. Final simplification 32.7%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023285 
(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))))