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

Percentage Accurate: 99.8% → 99.8%

Time: 8.5s

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, 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. Add Preprocessing

3. Final simplification 99.7%

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

Alternative 2: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-16}:\\ \;\;\;\;z \cdot \cos y + x \cdot y\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+69}:\\ \;\;\;\;x \cdot \sin y + z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.45e-16)
   (+ (* z (cos y)) (* x y))
   (if (<= z 1.35e+69) (+ (* x (sin y)) z) (fma (cos y) z (* x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.45e-16) {
		tmp = (z * cos(y)) + (x * y);
	} else if (z <= 1.35e+69) {
		tmp = (x * sin(y)) + z;
	} else {
		tmp = fma(cos(y), z, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.45e-16)
		tmp = Float64(Float64(z * cos(y)) + Float64(x * y));
	elseif (z <= 1.35e+69)
		tmp = Float64(Float64(x * sin(y)) + z);
	else
		tmp = fma(cos(y), z, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.45e-16], N[(N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.35e+69], N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4499999999999999e-16

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 81.2%

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

   if -1.4499999999999999e-16 < z < 1.3499999999999999e69

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 89.0%

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

   if 1.3499999999999999e69 < z

   1. Initial program 99.9%

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

      1. add-cube-cbrt 98.3%

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

      2. pow3 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in y around 0 80.5%

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

      1. +-commutative 80.5%

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

      2. unpow3 80.4%

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

      3. add-cube-cbrt 81.9%

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

      4. *-commutative 81.9%

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

      5. fma-define 81.9%

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

      6. *-commutative 81.9%

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

   7. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, y \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-16}:\\ \;\;\;\;z \cdot \cos y + x \cdot y\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+69}:\\ \;\;\;\;x \cdot \sin y + z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-15} \lor \neg \left(z \leq 6.5 \cdot 10^{+78}\right):\\ \;\;\;\;z \cdot \cos y + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.15e-15) (not (<= z 6.5e+78)))
   (+ (* z (cos y)) (* x y))
   (+ (* x (sin y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.15e-15) || !(z <= 6.5e+78)) {
		tmp = (z * cos(y)) + (x * y);
	} else {
		tmp = (x * 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 ((z <= (-1.15d-15)) .or. (.not. (z <= 6.5d+78))) then
        tmp = (z * cos(y)) + (x * y)
    else
        tmp = (x * sin(y)) + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.15e-15) || !(z <= 6.5e+78)) {
		tmp = (z * Math.cos(y)) + (x * y);
	} else {
		tmp = (x * Math.sin(y)) + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.15e-15) or not (z <= 6.5e+78):
		tmp = (z * math.cos(y)) + (x * y)
	else:
		tmp = (x * math.sin(y)) + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.15e-15) || !(z <= 6.5e+78))
		tmp = Float64(Float64(z * cos(y)) + Float64(x * y));
	else
		tmp = Float64(Float64(x * sin(y)) + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.15e-15) || ~((z <= 6.5e+78)))
		tmp = (z * cos(y)) + (x * y);
	else
		tmp = (x * sin(y)) + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.15e-15], N[Not[LessEqual[z, 6.5e+78]], $MachinePrecision]], N[(N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.14999999999999995e-15 or 6.50000000000000036e78 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 81.5%

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

   if -1.14999999999999995e-15 < z < 6.50000000000000036e78

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 89.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-15} \lor \neg \left(z \leq 6.5 \cdot 10^{+78}\right):\\ \;\;\;\;z \cdot \cos y + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y + z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.6% 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. Add Preprocessing

3. Taylor expanded in y around 0 75.6%

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

4. Final simplification 75.6%

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

Alternative 5: 41.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+92} \lor \neg \left(x \leq 1.7 \cdot 10^{+43}\right) \land \left(x \leq 7.4 \cdot 10^{+77} \lor \neg \left(x \leq 1.05 \cdot 10^{+162}\right)\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.05e+92)
         (and (not (<= x 1.7e+43)) (or (<= x 7.4e+77) (not (<= x 1.05e+162)))))
   (* x y)
   z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.05e+92) || (!(x <= 1.7e+43) && ((x <= 7.4e+77) || !(x <= 1.05e+162)))) {
		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 ((x <= (-2.05d+92)) .or. (.not. (x <= 1.7d+43)) .and. (x <= 7.4d+77) .or. (.not. (x <= 1.05d+162))) 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 ((x <= -2.05e+92) || (!(x <= 1.7e+43) && ((x <= 7.4e+77) || !(x <= 1.05e+162)))) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.05e+92) or (not (x <= 1.7e+43) and ((x <= 7.4e+77) or not (x <= 1.05e+162))):
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.05e+92) || (!(x <= 1.7e+43) && ((x <= 7.4e+77) || !(x <= 1.05e+162))))
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.05e+92) || (~((x <= 1.7e+43)) && ((x <= 7.4e+77) || ~((x <= 1.05e+162)))))
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.05e+92], And[N[Not[LessEqual[x, 1.7e+43]], $MachinePrecision], Or[LessEqual[x, 7.4e+77], N[Not[LessEqual[x, 1.05e+162]], $MachinePrecision]]]], N[(x * y), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -2.05000000000000012e92 or 1.70000000000000006e43 < x < 7.3999999999999999e77 or 1.05e162 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 66.6%

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

   4. Taylor expanded in y around 0 51.9%

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

      1. +-commutative 51.9%

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

   6. Simplified 51.9%

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

   7. Taylor expanded in x around inf 42.3%

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

      1. *-commutative 42.3%

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

   9. Simplified 42.3%

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

   if -2.05000000000000012e92 < x < 1.70000000000000006e43 or 7.3999999999999999e77 < x < 1.05e162

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 50.1%

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

   4. Taylor expanded in y around 0 44.6%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+92} \lor \neg \left(x \leq 1.7 \cdot 10^{+43}\right) \land \left(x \leq 7.4 \cdot 10^{+77} \lor \neg \left(x \leq 1.05 \cdot 10^{+162}\right)\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.7% 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. Add Preprocessing

3. Taylor expanded in y around 0 55.6%

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

4. Taylor expanded in y around 0 48.0%

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

   1. +-commutative 48.0%

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

6. Simplified 48.0%

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

7. Final simplification 48.0%

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

Alternative 7: 39.0% 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. Add Preprocessing

3. Taylor expanded in y around 0 55.6%

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

4. Taylor expanded in y around 0 33.3%

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

5. Final simplification 33.3%

   \[\leadsto z \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024048 
(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))))