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

Percentage Accurate: 99.8% → 99.8%

Time: 7.8s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 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%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 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. Final simplification 99.8%

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

Alternative 3: 75.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.002:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0115:\\ \;\;\;\;x + y \cdot \left(z + -0.5 \cdot \left(x \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+71} \lor \neg \left(y \leq 5.6 \cdot 10^{+248}\right) \land y \leq 2.4 \cdot 10^{+287}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.002)
     t_0
     (if (<= y 0.0115)
       (+ x (* y (+ z (* -0.5 (* x y)))))
       (if (or (<= y 1.05e+71) (and (not (<= y 5.6e+248)) (<= y 2.4e+287)))
         t_0
         (* z (sin y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.002) {
		tmp = t_0;
	} else if (y <= 0.0115) {
		tmp = x + (y * (z + (-0.5 * (x * y))));
	} else if ((y <= 1.05e+71) || (!(y <= 5.6e+248) && (y <= 2.4e+287))) {
		tmp = t_0;
	} else {
		tmp = z * 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) :: tmp
    t_0 = x * cos(y)
    if (y <= (-0.002d0)) then
        tmp = t_0
    else if (y <= 0.0115d0) then
        tmp = x + (y * (z + ((-0.5d0) * (x * y))))
    else if ((y <= 1.05d+71) .or. (.not. (y <= 5.6d+248)) .and. (y <= 2.4d+287)) then
        tmp = t_0
    else
        tmp = z * sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.cos(y);
	double tmp;
	if (y <= -0.002) {
		tmp = t_0;
	} else if (y <= 0.0115) {
		tmp = x + (y * (z + (-0.5 * (x * y))));
	} else if ((y <= 1.05e+71) || (!(y <= 5.6e+248) && (y <= 2.4e+287))) {
		tmp = t_0;
	} else {
		tmp = z * Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if y <= -0.002:
		tmp = t_0
	elif y <= 0.0115:
		tmp = x + (y * (z + (-0.5 * (x * y))))
	elif (y <= 1.05e+71) or (not (y <= 5.6e+248) and (y <= 2.4e+287)):
		tmp = t_0
	else:
		tmp = z * math.sin(y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.002)
		tmp = t_0;
	elseif (y <= 0.0115)
		tmp = Float64(x + Float64(y * Float64(z + Float64(-0.5 * Float64(x * y)))));
	elseif ((y <= 1.05e+71) || (!(y <= 5.6e+248) && (y <= 2.4e+287)))
		tmp = t_0;
	else
		tmp = Float64(z * sin(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (y <= -0.002)
		tmp = t_0;
	elseif (y <= 0.0115)
		tmp = x + (y * (z + (-0.5 * (x * y))));
	elseif ((y <= 1.05e+71) || (~((y <= 5.6e+248)) && (y <= 2.4e+287)))
		tmp = t_0;
	else
		tmp = z * sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.002], t$95$0, If[LessEqual[y, 0.0115], N[(x + N[(y * N[(z + N[(-0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 1.05e+71], And[N[Not[LessEqual[y, 5.6e+248]], $MachinePrecision], LessEqual[y, 2.4e+287]]], t$95$0, N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e-3 or 0.0115 < y < 1.04999999999999995e71 or 5.6000000000000004e248 < y < 2.3999999999999999e287

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 69.9%

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

   if -2e-3 < y < 0.0115

   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%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if 1.04999999999999995e71 < y < 5.6000000000000004e248 or 2.3999999999999999e287 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 72.6%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.002:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.0115:\\ \;\;\;\;x + y \cdot \left(z + -0.5 \cdot \left(x \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+71} \lor \neg \left(y \leq 5.6 \cdot 10^{+248}\right) \land y \leq 2.4 \cdot 10^{+287}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.002 \lor \neg \left(y \leq 0.0185\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + -0.5 \cdot \left(x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.002) (not (<= y 0.0185)))
   (* x (cos y))
   (+ x (* y (+ z (* -0.5 (* x y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.002) || !(y <= 0.0185)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * (z + (-0.5 * (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 <= (-0.002d0)) .or. (.not. (y <= 0.0185d0))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * (z + ((-0.5d0) * (x * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.002) || !(y <= 0.0185)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * (z + (-0.5 * (x * y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.002) or not (y <= 0.0185):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * (z + (-0.5 * (x * y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.002) || !(y <= 0.0185))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(z + Float64(-0.5 * Float64(x * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.002) || ~((y <= 0.0185)))
		tmp = x * cos(y);
	else
		tmp = x + (y * (z + (-0.5 * (x * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.002], N[Not[LessEqual[y, 0.0185]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z + N[(-0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2e-3 or 0.0184999999999999991 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 56.3%

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

   if -2e-3 < y < 0.0184999999999999991

   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%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.002 \lor \neg \left(y \leq 0.0185\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + -0.5 \cdot \left(x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.0% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+63}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 3e+63) x (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3e+63) {
		tmp = x;
	} else {
		tmp = 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 (z <= 3d+63) then
        tmp = x
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 3e+63) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 3e+63:
		tmp = x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3e+63)
		tmp = x;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 3e+63)
		tmp = x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3e+63], x, N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.99999999999999999e63

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf 86.5%

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

      1. associate-/l* 85.9%

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

   5. Simplified 85.9%

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

   6. Taylor expanded in y around 0 46.0%

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

   if 2.99999999999999999e63 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 70.7%

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

   4. Taylor expanded in y around 0 41.6%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+63}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.4% 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 54.4%

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

   1. +-commutative 54.4%

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

5. Simplified 54.4%

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

6. Final simplification 54.4%

   \[\leadsto x + y \cdot z \]
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. Taylor expanded in z around inf 89.2%

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

   1. associate-/l* 88.7%

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

5. Simplified 88.7%

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

6. Taylor expanded in y around 0 40.9%

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

7. Final simplification 40.9%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))