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

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. *-commutative 99.9%

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

   3. fma-def 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 4: 74.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+56}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-6} \lor \neg \left(y \leq 10^{-15}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -4.6e+113)
     t_0
     (if (<= y -2.7e+56)
       (* (sin y) z)
       (if (or (<= y -2.3e-6) (not (<= y 1e-15))) t_0 (+ x (* y z)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -4.6e+113) {
		tmp = t_0;
	} else if (y <= -2.7e+56) {
		tmp = sin(y) * z;
	} else if ((y <= -2.3e-6) || !(y <= 1e-15)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * cos(y)
    if (y <= (-4.6d+113)) then
        tmp = t_0
    else if (y <= (-2.7d+56)) then
        tmp = sin(y) * z
    else if ((y <= (-2.3d-6)) .or. (.not. (y <= 1d-15))) then
        tmp = t_0
    else
        tmp = x + (y * z)
    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 <= -4.6e+113) {
		tmp = t_0;
	} else if (y <= -2.7e+56) {
		tmp = Math.sin(y) * z;
	} else if ((y <= -2.3e-6) || !(y <= 1e-15)) {
		tmp = t_0;
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if y <= -4.6e+113:
		tmp = t_0
	elif y <= -2.7e+56:
		tmp = math.sin(y) * z
	elif (y <= -2.3e-6) or not (y <= 1e-15):
		tmp = t_0
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -4.6e+113)
		tmp = t_0;
	elseif (y <= -2.7e+56)
		tmp = Float64(sin(y) * z);
	elseif ((y <= -2.3e-6) || !(y <= 1e-15))
		tmp = t_0;
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (y <= -4.6e+113)
		tmp = t_0;
	elseif (y <= -2.7e+56)
		tmp = sin(y) * z;
	elseif ((y <= -2.3e-6) || ~((y <= 1e-15)))
		tmp = t_0;
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.6e+113], t$95$0, If[LessEqual[y, -2.7e+56], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision], If[Or[LessEqual[y, -2.3e-6], N[Not[LessEqual[y, 1e-15]], $MachinePrecision]], t$95$0, N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.59999999999999993e113 or -2.7000000000000001e56 < y < -2.3e-6 or 1.0000000000000001e-15 < y

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in z around 0 65.2%

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

   if -4.59999999999999993e113 < y < -2.7000000000000001e56

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 75.9%

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

   if -2.3e-6 < y < 1.0000000000000001e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.8%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+113}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+56}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-6} \lor \neg \left(y \leq 10^{-15}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 5: 74.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -1 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -8 \cdot 10^{+55}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-6} \lor \neg \left(y \leq 10^{-15}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -1e+113)
     t_0
     (if (<= y -8e+55)
       (* (sin y) z)
       (if (or (<= y -2.3e-6) (not (<= y 1e-15))) t_0 (fma y z x))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -1e+113) {
		tmp = t_0;
	} else if (y <= -8e+55) {
		tmp = sin(y) * z;
	} else if ((y <= -2.3e-6) || !(y <= 1e-15)) {
		tmp = t_0;
	} else {
		tmp = fma(y, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -1e+113)
		tmp = t_0;
	elseif (y <= -8e+55)
		tmp = Float64(sin(y) * z);
	elseif ((y <= -2.3e-6) || !(y <= 1e-15))
		tmp = t_0;
	else
		tmp = fma(y, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e+113], t$95$0, If[LessEqual[y, -8e+55], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision], If[Or[LessEqual[y, -2.3e-6], N[Not[LessEqual[y, 1e-15]], $MachinePrecision]], t$95$0, N[(y * z + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1e113 or -8.00000000000000008e55 < y < -2.3e-6 or 1.0000000000000001e-15 < y

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in z around 0 65.2%

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

   if -1e113 < y < -8.00000000000000008e55

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 75.9%

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

   if -2.3e-6 < y < 1.0000000000000001e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.8%

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

      1. fma-def 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+113}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -8 \cdot 10^{+55}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-6} \lor \neg \left(y \leq 10^{-15}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \]

Alternative 6: 86.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+16} \lor \neg \left(x \leq 4.8 \cdot 10^{+53}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + \sin y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.4e+16) (not (<= x 4.8e+53)))
   (* x (cos y))
   (+ x (* (sin y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.4e+16) || !(x <= 4.8e+53)) {
		tmp = x * cos(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 ((x <= (-2.4d+16)) .or. (.not. (x <= 4.8d+53))) then
        tmp = x * cos(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 ((x <= -2.4e+16) || !(x <= 4.8e+53)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (Math.sin(y) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.4e+16) or not (x <= 4.8e+53):
		tmp = x * math.cos(y)
	else:
		tmp = x + (math.sin(y) * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.4e+16) || !(x <= 4.8e+53))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(sin(y) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.4e+16) || ~((x <= 4.8e+53)))
		tmp = x * cos(y);
	else
		tmp = x + (sin(y) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.4e+16], N[Not[LessEqual[x, 4.8e+53]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e16 or 4.8e53 < x

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in z around 0 90.5%

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

   if -2.4e16 < x < 4.8e53

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 89.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+16} \lor \neg \left(x \leq 4.8 \cdot 10^{+53}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + \sin y \cdot z\\ \end{array} \]

Alternative 7: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00125 \lor \neg \left(y \leq 0.0017\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.00125) (not (<= y 0.0017))) (* (sin y) z) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00125) || !(y <= 0.0017)) {
		tmp = sin(y) * z;
	} 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 <= (-0.00125d0)) .or. (.not. (y <= 0.0017d0))) then
        tmp = sin(y) * z
    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 <= -0.00125) || !(y <= 0.0017)) {
		tmp = Math.sin(y) * z;
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.00125) or not (y <= 0.0017):
		tmp = math.sin(y) * z
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.00125) || !(y <= 0.0017))
		tmp = Float64(sin(y) * z);
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.00125) || ~((y <= 0.0017)))
		tmp = sin(y) * z;
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.00125], N[Not[LessEqual[y, 0.0017]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00125000000000000003 or 0.00169999999999999991 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 41.3%

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

   if -0.00125000000000000003 < y < 0.00169999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.3%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00125 \lor \neg \left(y \leq 0.0017\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 8: 41.3% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-197}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-247}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-108}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-52}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.22e-197)
   x
   (if (<= x 2.75e-247)
     (* y z)
     (if (<= x 2.8e-108) x (if (<= x 1.7e-52) (* y z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.22e-197) {
		tmp = x;
	} else if (x <= 2.75e-247) {
		tmp = y * z;
	} else if (x <= 2.8e-108) {
		tmp = x;
	} else if (x <= 1.7e-52) {
		tmp = y * z;
	} else {
		tmp = 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 (x <= (-1.22d-197)) then
        tmp = x
    else if (x <= 2.75d-247) then
        tmp = y * z
    else if (x <= 2.8d-108) then
        tmp = x
    else if (x <= 1.7d-52) then
        tmp = y * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.22e-197) {
		tmp = x;
	} else if (x <= 2.75e-247) {
		tmp = y * z;
	} else if (x <= 2.8e-108) {
		tmp = x;
	} else if (x <= 1.7e-52) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.22e-197:
		tmp = x
	elif x <= 2.75e-247:
		tmp = y * z
	elif x <= 2.8e-108:
		tmp = x
	elif x <= 1.7e-52:
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.22e-197)
		tmp = x;
	elseif (x <= 2.75e-247)
		tmp = Float64(y * z);
	elseif (x <= 2.8e-108)
		tmp = x;
	elseif (x <= 1.7e-52)
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.22e-197)
		tmp = x;
	elseif (x <= 2.75e-247)
		tmp = y * z;
	elseif (x <= 2.8e-108)
		tmp = x;
	elseif (x <= 1.7e-52)
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.22e-197], x, If[LessEqual[x, 2.75e-247], N[(y * z), $MachinePrecision], If[LessEqual[x, 2.8e-108], x, If[LessEqual[x, 1.7e-52], N[(y * z), $MachinePrecision], x]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.22e-197 or 2.74999999999999997e-247 < x < 2.8e-108 or 1.70000000000000009e-52 < x

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 47.6%

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

   if -1.22e-197 < x < 2.74999999999999997e-247 or 2.8e-108 < x < 1.70000000000000009e-52

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 75.7%

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

   3. Taylor expanded in y around inf 57.4%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-197}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-247}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-108}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-52}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 52.3% 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 99.9%

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

2. Taylor expanded in y around 0 57.6%

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

3. Final simplification 57.6%

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

Alternative 10: 39.5% 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 99.9%

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

   1. +-commutative 99.9%

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

   2. *-commutative 99.9%

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

   3. fma-def 99.9%

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

3. Applied egg-rr 99.9%

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

4. Taylor expanded in y around 0 43.6%

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

5. Final simplification 43.6%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023238 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))