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

Percentage Accurate: 99.8% → 99.8%

Time: 8.6s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. fma-neg 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 73.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+71} \lor \neg \left(z \leq 1.05 \cdot 10^{+44} \lor \neg \left(z \leq 2.7 \cdot 10^{+145}\right) \land z \leq 3.2 \cdot 10^{+185}\right):\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.06e+71)
         (not
          (or (<= z 1.05e+44) (and (not (<= z 2.7e+145)) (<= z 3.2e+185)))))
   (* z (- (sin y)))
   (* (cos y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.06e+71) || !((z <= 1.05e+44) || (!(z <= 2.7e+145) && (z <= 3.2e+185)))) {
		tmp = z * -sin(y);
	} else {
		tmp = cos(y) * 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 ((z <= (-1.06d+71)) .or. (.not. (z <= 1.05d+44) .or. (.not. (z <= 2.7d+145)) .and. (z <= 3.2d+185))) then
        tmp = z * -sin(y)
    else
        tmp = cos(y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.06e+71) || !((z <= 1.05e+44) || (!(z <= 2.7e+145) && (z <= 3.2e+185)))) {
		tmp = z * -Math.sin(y);
	} else {
		tmp = Math.cos(y) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.06e+71) or not ((z <= 1.05e+44) or (not (z <= 2.7e+145) and (z <= 3.2e+185))):
		tmp = z * -math.sin(y)
	else:
		tmp = math.cos(y) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.06e+71) || !((z <= 1.05e+44) || (!(z <= 2.7e+145) && (z <= 3.2e+185))))
		tmp = Float64(z * Float64(-sin(y)));
	else
		tmp = Float64(cos(y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.06e+71) || ~(((z <= 1.05e+44) || (~((z <= 2.7e+145)) && (z <= 3.2e+185)))))
		tmp = z * -sin(y);
	else
		tmp = cos(y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.06e+71], N[Not[Or[LessEqual[z, 1.05e+44], And[N[Not[LessEqual[z, 2.7e+145]], $MachinePrecision], LessEqual[z, 3.2e+185]]]], $MachinePrecision]], N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.06e71 or 1.04999999999999993e44 < z < 2.70000000000000022e145 or 3.20000000000000006e185 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 78.1%

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

      1. mul-1-neg 78.1%

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

      2. *-commutative 78.1%

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

      3. distribute-rgt-neg-in 78.1%

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

   5. Simplified 78.1%

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

   if -1.06e71 < z < 1.04999999999999993e44 or 2.70000000000000022e145 < z < 3.20000000000000006e185

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 85.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+71} \lor \neg \left(z \leq 1.05 \cdot 10^{+44} \lor \neg \left(z \leq 2.7 \cdot 10^{+145}\right) \land z \leq 3.2 \cdot 10^{+185}\right):\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-16} \lor \neg \left(z \leq 1.1 \cdot 10^{+35}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8.5e-16) (not (<= z 1.1e+35)))
   (- x (* z (sin y)))
   (* (cos y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.5e-16) || !(z <= 1.1e+35)) {
		tmp = x - (z * sin(y));
	} else {
		tmp = cos(y) * 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 ((z <= (-8.5d-16)) .or. (.not. (z <= 1.1d+35))) then
        tmp = x - (z * sin(y))
    else
        tmp = cos(y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.5e-16) || !(z <= 1.1e+35)) {
		tmp = x - (z * Math.sin(y));
	} else {
		tmp = Math.cos(y) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8.5e-16) or not (z <= 1.1e+35):
		tmp = x - (z * math.sin(y))
	else:
		tmp = math.cos(y) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8.5e-16) || !(z <= 1.1e+35))
		tmp = Float64(x - Float64(z * sin(y)));
	else
		tmp = Float64(cos(y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8.5e-16) || ~((z <= 1.1e+35)))
		tmp = x - (z * sin(y));
	else
		tmp = cos(y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8.5e-16], N[Not[LessEqual[z, 1.1e+35]], $MachinePrecision]], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5000000000000001e-16 or 1.0999999999999999e35 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 93.7%

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

   if -8.5000000000000001e-16 < z < 1.0999999999999999e35

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 91.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-16} \lor \neg \left(z \leq 1.1 \cdot 10^{+35}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -102000 \lor \neg \left(y \leq 34\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -102000.0) (not (<= y 34.0))) (* (cos y) x) (- x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -102000.0) || !(y <= 34.0)) {
		tmp = cos(y) * x;
	} 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 <= (-102000.0d0)) .or. (.not. (y <= 34.0d0))) then
        tmp = cos(y) * x
    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 <= -102000.0) || !(y <= 34.0)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = x - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -102000.0) or not (y <= 34.0):
		tmp = math.cos(y) * x
	else:
		tmp = x - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -102000.0) || !(y <= 34.0))
		tmp = Float64(cos(y) * x);
	else
		tmp = Float64(x - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -102000.0) || ~((y <= 34.0)))
		tmp = cos(y) * x;
	else
		tmp = x - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -102000.0], N[Not[LessEqual[y, 34.0]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -102000 or 34 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 50.3%

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

   if -102000 < y < 34

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 97.3%

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

      1. mul-1-neg 97.3%

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

      2. unsub-neg 97.3%

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

   5. Simplified 97.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -102000 \lor \neg \left(y \leq 34\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.0% 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 53.4%

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

   1. mul-1-neg 53.4%

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

   2. unsub-neg 53.4%

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

5. Simplified 53.4%

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

6. Final simplification 53.4%

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

Alternative 7: 39.1% 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%

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

   2. fma-neg 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in y around 0 39.7%

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

6. Final simplification 39.7%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (- (* x (cos y)) (* z (sin y))))