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

Percentage Accurate: 99.8% → 99.8%

Time: 9.4s

Alternatives: 8

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, 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]

Derivation

1. Initial program 99.8%

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

Alternative 2: 79.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 + z \cdot \frac{\sin y}{x}\right)\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+189}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ 1.0 (* z (/ (sin y) x))))))
   (if (<= z -2.9e-12)
     t_0
     (if (<= z 2.7e-29)
       (* x (cos y))
       (if (<= z 5.6e+189) t_0 (* z (sin y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (1.0 + (z * (sin(y) / x)));
	double tmp;
	if (z <= -2.9e-12) {
		tmp = t_0;
	} else if (z <= 2.7e-29) {
		tmp = x * cos(y);
	} else if (z <= 5.6e+189) {
		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 * (1.0d0 + (z * (sin(y) / x)))
    if (z <= (-2.9d-12)) then
        tmp = t_0
    else if (z <= 2.7d-29) then
        tmp = x * cos(y)
    else if (z <= 5.6d+189) 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 * (1.0 + (z * (Math.sin(y) / x)));
	double tmp;
	if (z <= -2.9e-12) {
		tmp = t_0;
	} else if (z <= 2.7e-29) {
		tmp = x * Math.cos(y);
	} else if (z <= 5.6e+189) {
		tmp = t_0;
	} else {
		tmp = z * Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (1.0 + (z * (math.sin(y) / x)))
	tmp = 0
	if z <= -2.9e-12:
		tmp = t_0
	elif z <= 2.7e-29:
		tmp = x * math.cos(y)
	elif z <= 5.6e+189:
		tmp = t_0
	else:
		tmp = z * math.sin(y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(1.0 + Float64(z * Float64(sin(y) / x))))
	tmp = 0.0
	if (z <= -2.9e-12)
		tmp = t_0;
	elseif (z <= 2.7e-29)
		tmp = Float64(x * cos(y));
	elseif (z <= 5.6e+189)
		tmp = t_0;
	else
		tmp = Float64(z * sin(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (1.0 + (z * (sin(y) / x)));
	tmp = 0.0;
	if (z <= -2.9e-12)
		tmp = t_0;
	elseif (z <= 2.7e-29)
		tmp = x * cos(y);
	elseif (z <= 5.6e+189)
		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[(1.0 + N[(z * N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.9e-12], t$95$0, If[LessEqual[z, 2.7e-29], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.6e+189], t$95$0, N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.9000000000000002e-12 or 2.70000000000000023e-29 < z < 5.60000000000000013e189

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 90.8%

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

      1. associate-/l* 90.7%

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

   5. Simplified 90.7%

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

   6. Taylor expanded in y around 0 77.9%

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

   if -2.9000000000000002e-12 < z < 2.70000000000000023e-29

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 87.5%

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

   if 5.60000000000000013e189 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.3%

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

Alternative 3: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.019:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.041:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+65}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.019)
     t_0
     (if (<= y 0.041)
       (+ x (* y (+ z (* y (+ (* x -0.5) (* -0.16666666666666666 (* y z)))))))
       (if (<= y 8e+65) (* z (sin y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.019) {
		tmp = t_0;
	} else if (y <= 0.041) {
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))));
	} else if (y <= 8e+65) {
		tmp = z * sin(y);
	} else {
		tmp = t_0;
	}
	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 <= (-0.019d0)) then
        tmp = t_0
    else if (y <= 0.041d0) then
        tmp = x + (y * (z + (y * ((x * (-0.5d0)) + ((-0.16666666666666666d0) * (y * z))))))
    else if (y <= 8d+65) then
        tmp = z * sin(y)
    else
        tmp = t_0
    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 <= -0.019) {
		tmp = t_0;
	} else if (y <= 0.041) {
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))));
	} else if (y <= 8e+65) {
		tmp = z * Math.sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if y <= -0.019:
		tmp = t_0
	elif y <= 0.041:
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))))
	elif y <= 8e+65:
		tmp = z * math.sin(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.019)
		tmp = t_0;
	elseif (y <= 0.041)
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * Float64(Float64(x * -0.5) + Float64(-0.16666666666666666 * Float64(y * z)))))));
	elseif (y <= 8e+65)
		tmp = Float64(z * sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (y <= -0.019)
		tmp = t_0;
	elseif (y <= 0.041)
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))));
	elseif (y <= 8e+65)
		tmp = z * sin(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.019], t$95$0, If[LessEqual[y, 0.041], N[(x + N[(y * N[(z + N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(-0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+65], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0189999999999999995 or 7.9999999999999999e65 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 61.1%

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

   if -0.0189999999999999995 < y < 0.0410000000000000017

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.7%

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

   if 0.0410000000000000017 < y < 7.9999999999999999e65

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 66.1%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.019:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.041:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+65}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.02 \lor \neg \left(y \leq 4.5 \cdot 10^{-24}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.02) (not (<= y 4.5e-24)))
   (* x (cos y))
   (+ x (* y (+ z (* y (+ (* x -0.5) (* -0.16666666666666666 (* y z)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.02) || !(y <= 4.5e-24)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (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 <= (-0.02d0)) .or. (.not. (y <= 4.5d-24))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * (z + (y * ((x * (-0.5d0)) + ((-0.16666666666666666d0) * (y * z))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.02) || !(y <= 4.5e-24)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.02) or not (y <= 4.5e-24):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.02) || !(y <= 4.5e-24))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * Float64(Float64(x * -0.5) + Float64(-0.16666666666666666 * Float64(y * z)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.02) || ~((y <= 4.5e-24)))
		tmp = x * cos(y);
	else
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.02], N[Not[LessEqual[y, 4.5e-24]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z + N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(-0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0200000000000000004 or 4.4999999999999997e-24 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 59.3%

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

   if -0.0200000000000000004 < y < 4.4999999999999997e-24

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.02 \lor \neg \left(y \leq 4.5 \cdot 10^{-24}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.2% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{+202}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 3.4e+202) x (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.4e+202) {
		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 <= 3.4d+202) 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 <= 3.4e+202) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 3.4e+202:
		tmp = x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.4e+202)
		tmp = x;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 3.4e+202)
		tmp = x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3.4e+202], x, N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.4e202

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 53.1%

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

   4. Taylor expanded in x around -inf 51.8%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot z}{x} - 1\right)\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 51.8%

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

      2. *-commutative 51.8%

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

      3. distribute-rgt-neg-in 51.8%

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

      4. sub-neg 51.8%

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

      5. metadata-eval 51.8%

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

      6. +-commutative 51.8%

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

      7. mul-1-neg 51.8%

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

      8. unsub-neg 51.8%

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

      9. *-commutative 51.8%

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

      10. associate-/l* 51.6%

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

   6. Simplified 51.6%

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

   7. Taylor expanded in z around 0 43.0%

      \[\leadsto \color{blue}{-1} \cdot \left(-x\right) \]
   8. Step-by-step derivation

      1. mul-1-neg 43.0%

         \[\leadsto \color{blue}{-\left(-x\right)} \]

      2. neg-sub0 43.0%

         \[\leadsto \color{blue}{0 - \left(-x\right)} \]

      3. add-sqr-sqrt 20.5%

         \[\leadsto 0 - \color{blue}{\sqrt{-x} \cdot \sqrt{-x}} \]

      4. sqrt-unprod 14.4%

         \[\leadsto 0 - \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \]

      5. sqr-neg 14.4%

         \[\leadsto 0 - \sqrt{\color{blue}{x \cdot x}} \]

      6. sqrt-unprod 2.3%

         \[\leadsto 0 - \color{blue}{\sqrt{x} \cdot \sqrt{x}} \]

      7. add-sqr-sqrt 4.9%

         \[\leadsto 0 - \color{blue}{x} \]

      8. +-lft-identity 4.9%

         \[\leadsto 0 - \color{blue}{\left(0 + x\right)} \]

      9. +-commutative 4.9%

         \[\leadsto 0 - \color{blue}{\left(x + 0\right)} \]

      10. associate--r+ 4.9%

          \[\leadsto \color{blue}{\left(0 - x\right) - 0} \]

      11. neg-sub0 4.9%

          \[\leadsto \color{blue}{\left(-x\right)} - 0 \]

      12. add-sqr-sqrt 2.6%

          \[\leadsto \color{blue}{\sqrt{-x} \cdot \sqrt{-x}} - 0 \]

      13. sqrt-unprod 15.1%

          \[\leadsto \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} - 0 \]

      14. sqr-neg 15.1%

          \[\leadsto \sqrt{\color{blue}{x \cdot x}} - 0 \]

      15. sqrt-unprod 22.2%

          \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{x}} - 0 \]

      16. add-sqr-sqrt 43.0%

          \[\leadsto \color{blue}{x} - 0 \]

   9. Applied egg-rr 43.0%

      \[\leadsto \color{blue}{x - 0} \]

   if 3.4e202 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 80.4%

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

   4. Taylor expanded in y around 0 42.6%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{+202}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.6% 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 53.2%

   \[\leadsto \color{blue}{x + y \cdot z} \]
4. Add Preprocessing

Alternative 7: 17.2% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 37.1%

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

4. Taylor expanded in y around 0 16.9%

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

5. Final simplification 16.9%

   \[\leadsto y \cdot z \]
6. Add Preprocessing

Alternative 8: 4.9% accurate, 103.5× 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 Float64(-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 y around 0 53.2%

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

4. Taylor expanded in x around -inf 51.7%

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot z}{x} - 1\right)\right)} \]
5. Step-by-step derivation

   1. mul-1-neg 51.7%

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

   2. *-commutative 51.7%

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

   3. distribute-rgt-neg-in 51.7%

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

   4. sub-neg 51.7%

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

   5. metadata-eval 51.7%

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

   6. +-commutative 51.7%

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

   7. mul-1-neg 51.7%

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

   8. unsub-neg 51.7%

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

   9. *-commutative 51.7%

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

   10. associate-/l* 51.5%

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

6. Simplified 51.5%

   \[\leadsto \color{blue}{\left(-1 - z \cdot \frac{y}{x}\right) \cdot \left(-x\right)} \]
7. Step-by-step derivation

   1. neg-sub0 51.5%

      \[\leadsto \left(-1 - z \cdot \frac{y}{x}\right) \cdot \color{blue}{\left(0 - x\right)} \]

   2. sub-neg 51.5%

      \[\leadsto \left(-1 - z \cdot \frac{y}{x}\right) \cdot \color{blue}{\left(0 + \left(-x\right)\right)} \]

   3. add-sqr-sqrt 25.3%

      \[\leadsto \left(-1 - z \cdot \frac{y}{x}\right) \cdot \left(0 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right) \]

   4. sqrt-unprod 17.1%

      \[\leadsto \left(-1 - z \cdot \frac{y}{x}\right) \cdot \left(0 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right) \]

   5. sqr-neg 17.1%

      \[\leadsto \left(-1 - z \cdot \frac{y}{x}\right) \cdot \left(0 + \sqrt{\color{blue}{x \cdot x}}\right) \]

   6. sqrt-unprod 1.7%

      \[\leadsto \left(-1 - z \cdot \frac{y}{x}\right) \cdot \left(0 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) \]

   7. add-sqr-sqrt 3.3%

      \[\leadsto \left(-1 - z \cdot \frac{y}{x}\right) \cdot \left(0 + \color{blue}{x}\right) \]

8. Applied egg-rr 3.3%

   \[\leadsto \left(-1 - z \cdot \frac{y}{x}\right) \cdot \color{blue}{\left(0 + x\right)} \]

9. Taylor expanded in z around 0 4.7%

   \[\leadsto \color{blue}{-1 \cdot x} \]
10. Step-by-step derivation

    1. neg-mul-1 4.7%

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

11. Simplified 4.7%

    \[\leadsto \color{blue}{-x} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))