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

Percentage Accurate: 99.8% → 99.8%

Time: 9.3s

Alternatives: 8

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 75.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ \mathbf{if}\;y \leq -0.002:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.027:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5 + -0.16666666666666666 \cdot \left(x \cdot y\right)\right)\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+71} \lor \neg \left(y \leq 3 \cdot 10^{+246}\right) \land y \leq 9.2 \cdot 10^{+294}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))))
   (if (<= y -0.002)
     t_0
     (if (<= y 0.027)
       (+ z (* y (+ x (* y (+ (* z -0.5) (* -0.16666666666666666 (* x y)))))))
       (if (or (<= y 3.1e+71) (and (not (<= y 3e+246)) (<= y 9.2e+294)))
         t_0
         (* z (cos y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double tmp;
	if (y <= -0.002) {
		tmp = t_0;
	} else if (y <= 0.027) {
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))));
	} else if ((y <= 3.1e+71) || (!(y <= 3e+246) && (y <= 9.2e+294))) {
		tmp = t_0;
	} else {
		tmp = z * cos(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 * sin(y)
    if (y <= (-0.002d0)) then
        tmp = t_0
    else if (y <= 0.027d0) then
        tmp = z + (y * (x + (y * ((z * (-0.5d0)) + ((-0.16666666666666666d0) * (x * y))))))
    else if ((y <= 3.1d+71) .or. (.not. (y <= 3d+246)) .and. (y <= 9.2d+294)) then
        tmp = t_0
    else
        tmp = z * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.sin(y);
	double tmp;
	if (y <= -0.002) {
		tmp = t_0;
	} else if (y <= 0.027) {
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))));
	} else if ((y <= 3.1e+71) || (!(y <= 3e+246) && (y <= 9.2e+294))) {
		tmp = t_0;
	} else {
		tmp = z * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.sin(y)
	tmp = 0
	if y <= -0.002:
		tmp = t_0
	elif y <= 0.027:
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))))
	elif (y <= 3.1e+71) or (not (y <= 3e+246) and (y <= 9.2e+294)):
		tmp = t_0
	else:
		tmp = z * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	tmp = 0.0
	if (y <= -0.002)
		tmp = t_0;
	elseif (y <= 0.027)
		tmp = Float64(z + Float64(y * Float64(x + Float64(y * Float64(Float64(z * -0.5) + Float64(-0.16666666666666666 * Float64(x * y)))))));
	elseif ((y <= 3.1e+71) || (!(y <= 3e+246) && (y <= 9.2e+294)))
		tmp = t_0;
	else
		tmp = Float64(z * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * sin(y);
	tmp = 0.0;
	if (y <= -0.002)
		tmp = t_0;
	elseif (y <= 0.027)
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))));
	elseif ((y <= 3.1e+71) || (~((y <= 3e+246)) && (y <= 9.2e+294)))
		tmp = t_0;
	else
		tmp = z * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.002], t$95$0, If[LessEqual[y, 0.027], N[(z + N[(y * N[(x + N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(-0.16666666666666666 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 3.1e+71], And[N[Not[LessEqual[y, 3e+246]], $MachinePrecision], LessEqual[y, 9.2e+294]]], t$95$0, N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e-3 or 0.0269999999999999997 < y < 3.10000000000000018e71 or 3e246 < y < 9.19999999999999987e294

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 70.5%

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

   if -2e-3 < y < 0.0269999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{z + y \cdot \left(x + y \cdot \left(-0.5 \cdot z + -0.16666666666666666 \cdot \left(x \cdot y\right)\right)\right)} \]

   if 3.10000000000000018e71 < y < 3e246 or 9.19999999999999987e294 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 72.7%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.002:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq 0.027:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5 + -0.16666666666666666 \cdot \left(x \cdot y\right)\right)\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+71} \lor \neg \left(y \leq 3 \cdot 10^{+246}\right) \land y \leq 9.2 \cdot 10^{+294}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ \mathbf{if}\;y \leq -2.95 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.00355:\\ \;\;\;\;\mathsf{fma}\left(x, y, z\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+73} \lor \neg \left(y \leq 4.6 \cdot 10^{+247}\right) \land y \leq 7.2 \cdot 10^{+286}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))))
   (if (<= y -2.95e-7)
     t_0
     (if (<= y 0.00355)
       (fma x y z)
       (if (or (<= y 2.15e+73) (and (not (<= y 4.6e+247)) (<= y 7.2e+286)))
         t_0
         (* z (cos y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double tmp;
	if (y <= -2.95e-7) {
		tmp = t_0;
	} else if (y <= 0.00355) {
		tmp = fma(x, y, z);
	} else if ((y <= 2.15e+73) || (!(y <= 4.6e+247) && (y <= 7.2e+286))) {
		tmp = t_0;
	} else {
		tmp = z * cos(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	tmp = 0.0
	if (y <= -2.95e-7)
		tmp = t_0;
	elseif (y <= 0.00355)
		tmp = fma(x, y, z);
	elseif ((y <= 2.15e+73) || (!(y <= 4.6e+247) && (y <= 7.2e+286)))
		tmp = t_0;
	else
		tmp = Float64(z * cos(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.95e-7], t$95$0, If[LessEqual[y, 0.00355], N[(x * y + z), $MachinePrecision], If[Or[LessEqual[y, 2.15e+73], And[N[Not[LessEqual[y, 4.6e+247]], $MachinePrecision], LessEqual[y, 7.2e+286]]], t$95$0, N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.94999999999999981e-7 or 0.0035500000000000002 < y < 2.15000000000000007e73 or 4.59999999999999981e247 < y < 7.2000000000000001e286

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 70.8%

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

   if -2.94999999999999981e-7 < y < 0.0035500000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   5. Simplified 100.0%

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

   if 2.15000000000000007e73 < y < 4.59999999999999981e247 or 7.2000000000000001e286 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 72.7%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq 0.00355:\\ \;\;\;\;\mathsf{fma}\left(x, y, z\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+73} \lor \neg \left(y \leq 4.6 \cdot 10^{+247}\right) \land y \leq 7.2 \cdot 10^{+286}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.002 \lor \neg \left(y \leq 0.105\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5 + -0.16666666666666666 \cdot \left(x \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.002) (not (<= y 0.105)))
   (* x (sin y))
   (+ z (* y (+ x (* y (+ (* z -0.5) (* -0.16666666666666666 (* x y)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.002) || !(y <= 0.105)) {
		tmp = x * sin(y);
	} else {
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (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 <= 0.105d0))) then
        tmp = x * sin(y)
    else
        tmp = z + (y * (x + (y * ((z * (-0.5d0)) + ((-0.16666666666666666d0) * (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 <= 0.105)) {
		tmp = x * Math.sin(y);
	} else {
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.002) or not (y <= 0.105):
		tmp = x * math.sin(y)
	else:
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.002) || !(y <= 0.105))
		tmp = Float64(x * sin(y));
	else
		tmp = Float64(z + Float64(y * Float64(x + Float64(y * Float64(Float64(z * -0.5) + Float64(-0.16666666666666666 * Float64(x * y)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.002) || ~((y <= 0.105)))
		tmp = x * sin(y);
	else
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.002], N[Not[LessEqual[y, 0.105]], $MachinePrecision]], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(z + N[(y * N[(x + N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(-0.16666666666666666 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2e-3 or 0.104999999999999996 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 56.7%

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

   if -2e-3 < y < 0.104999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{z + y \cdot \left(x + y \cdot \left(-0.5 \cdot z + -0.16666666666666666 \cdot \left(x \cdot y\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.002 \lor \neg \left(y \leq 0.105\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5 + -0.16666666666666666 \cdot \left(x \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.6% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+133} \lor \neg \left(x \leq 2.7 \cdot 10^{+188}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.2e+133) (not (<= x 2.7e+188))) (* x y) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.2e+133) || !(x <= 2.7e+188)) {
		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 <= (-5.2d+133)) .or. (.not. (x <= 2.7d+188))) 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 <= -5.2e+133) || !(x <= 2.7e+188)) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.2e+133) or not (x <= 2.7e+188):
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.2e+133) || !(x <= 2.7e+188))
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.2e+133) || ~((x <= 2.7e+188)))
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.2e+133], N[Not[LessEqual[x, 2.7e+188]], $MachinePrecision]], N[(x * y), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -5.1999999999999995e133 or 2.7e188 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 63.5%

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

      1. +-commutative 63.5%

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

   5. Simplified 63.5%

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

   6. Taylor expanded in x around inf 42.4%

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

      1. *-commutative 42.4%

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

   8. Simplified 42.4%

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

   if -5.1999999999999995e133 < x < 2.7e188

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. add-cube-cbrt 99.2%

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

      4. associate-*l* 99.2%

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

      5. fma-define 99.2%

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

      6. pow2 99.2%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\cos y}\right)}^{2}}, \sqrt[3]{\cos y} \cdot z, x \cdot \sin y\right) \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\cos y}\right)}^{2}, \sqrt[3]{\cos y} \cdot z, x \cdot \sin y\right)} \]

   5. Taylor expanded in y around 0 43.9%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+133} \lor \neg \left(x \leq 2.7 \cdot 10^{+188}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.2% 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 99.8%

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

3. Taylor expanded in y around 0 54.1%

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

   1. +-commutative 54.1%

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

5. Simplified 54.1%

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

6. Final simplification 54.1%

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

Alternative 8: 38.8% 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 99.8%

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

   1. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. add-cube-cbrt 99.4%

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

   4. associate-*l* 99.4%

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

   5. fma-define 99.4%

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

   6. pow2 99.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\cos y}\right)}^{2}}, \sqrt[3]{\cos y} \cdot z, x \cdot \sin y\right) \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\cos y}\right)}^{2}, \sqrt[3]{\cos y} \cdot z, x \cdot \sin y\right)} \]

5. Taylor expanded in y around 0 38.3%

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

6. Final simplification 38.3%

   \[\leadsto z \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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))))