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

Percentage Accurate: 99.8% → 99.8%

Time: 7.2s

Alternatives: 6

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, 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]

Derivation

1. Initial program 99.7%

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

2. Final simplification 99.7%

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

Alternative 2: 82.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.76 \cdot 10^{-172} \lor \neg \left(x \leq 4.7 \cdot 10^{-76}\right):\\ \;\;\;\;x \cdot \sin y + z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.76e-172) (not (<= x 4.7e-76)))
   (+ (* x (sin y)) z)
   (+ (* z (cos y)) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.76e-172) || !(x <= 4.7e-76)) {
		tmp = (x * sin(y)) + z;
	} else {
		tmp = (z * cos(y)) + (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 ((x <= (-1.76d-172)) .or. (.not. (x <= 4.7d-76))) then
        tmp = (x * sin(y)) + z
    else
        tmp = (z * cos(y)) + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.76e-172) || !(x <= 4.7e-76)) {
		tmp = (x * Math.sin(y)) + z;
	} else {
		tmp = (z * Math.cos(y)) + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.76e-172) or not (x <= 4.7e-76):
		tmp = (x * math.sin(y)) + z
	else:
		tmp = (z * math.cos(y)) + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.76e-172) || !(x <= 4.7e-76))
		tmp = Float64(Float64(x * sin(y)) + z);
	else
		tmp = Float64(Float64(z * cos(y)) + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.76e-172) || ~((x <= 4.7e-76)))
		tmp = (x * sin(y)) + z;
	else
		tmp = (z * cos(y)) + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.76e-172], N[Not[LessEqual[x, 4.7e-76]], $MachinePrecision]], N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision], N[(N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.76e-172 or 4.7000000000000002e-76 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 85.2%

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

   if -1.76e-172 < x < 4.7000000000000002e-76

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 81.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.76 \cdot 10^{-172} \lor \neg \left(x \leq 4.7 \cdot 10^{-76}\right):\\ \;\;\;\;x \cdot \sin y + z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y + x \cdot y\\ \end{array} \]

Alternative 3: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.216 \lor \neg \left(y \leq 5 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot \left(1 + -0.5 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.216) (not (<= y 5e-13)))
   (* x (sin y))
   (+ (* x y) (* z (+ 1.0 (* -0.5 (* y y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.216) || !(y <= 5e-13)) {
		tmp = x * sin(y);
	} else {
		tmp = (x * y) + (z * (1.0 + (-0.5 * (y * 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 <= (-0.216d0)) .or. (.not. (y <= 5d-13))) then
        tmp = x * sin(y)
    else
        tmp = (x * y) + (z * (1.0d0 + ((-0.5d0) * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.216) || !(y <= 5e-13)) {
		tmp = x * Math.sin(y);
	} else {
		tmp = (x * y) + (z * (1.0 + (-0.5 * (y * y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.216) or not (y <= 5e-13):
		tmp = x * math.sin(y)
	else:
		tmp = (x * y) + (z * (1.0 + (-0.5 * (y * y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.216) || !(y <= 5e-13))
		tmp = Float64(x * sin(y));
	else
		tmp = Float64(Float64(x * y) + Float64(z * Float64(1.0 + Float64(-0.5 * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.216) || ~((y <= 5e-13)))
		tmp = x * sin(y);
	else
		tmp = (x * y) + (z * (1.0 + (-0.5 * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.216], N[Not[LessEqual[y, 5e-13]], $MachinePrecision]], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] + N[(z * N[(1.0 + N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.215999999999999998 or 4.9999999999999999e-13 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. add-sqr-sqrt 57.9%

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

      4. associate-*r* 57.8%

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

      5. fma-def 57.8%

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

   3. Applied egg-rr 57.8%

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

   4. Taylor expanded in z around 0 56.7%

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

   if -0.215999999999999998 < y < 4.9999999999999999e-13

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around 0 99.9%

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

   5. Taylor expanded in y around 0 99.4%

      \[\leadsto y \cdot x + z \cdot \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 99.4%

         \[\leadsto y \cdot x + z \cdot \left(1 + -0.5 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   7. Simplified 99.4%

      \[\leadsto y \cdot x + z \cdot \color{blue}{\left(1 + -0.5 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.216 \lor \neg \left(y \leq 5 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot \left(1 + -0.5 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 4: 76.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in y around 0 76.5%

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

3. Final simplification 76.5%

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

Alternative 5: 52.5% 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%

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

3. Final simplification 46.2%

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

Alternative 6: 16.8% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. add-log-exp 99.6%

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

3. Applied egg-rr 99.6%

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

4. Taylor expanded in y around 0 58.1%

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

5. Taylor expanded in y around inf 15.3%

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

6. Final simplification 15.3%

   \[\leadsto x \cdot y \]

Reproduce

herbie shell --seed 2023207 
(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))))