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

Percentage Accurate: 99.8% → 99.8%

Time: 8.0s

Alternatives: 9

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 85.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+62} \lor \neg \left(x \leq 8.5 \cdot 10^{-59}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.35e+62) (not (<= x 8.5e-59)))
   (* x (cos y))
   (+ x (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.35e+62) || !(x <= 8.5e-59)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (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) :: tmp
    if ((x <= (-2.35d+62)) .or. (.not. (x <= 8.5d-59))) then
        tmp = x * cos(y)
    else
        tmp = x + (z * sin(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.35e+62) || !(x <= 8.5e-59)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.35e+62) or not (x <= 8.5e-59):
		tmp = x * math.cos(y)
	else:
		tmp = x + (z * math.sin(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.35e+62) || !(x <= 8.5e-59))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(z * sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.35e+62) || ~((x <= 8.5e-59)))
		tmp = x * cos(y);
	else
		tmp = x + (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.35e+62], N[Not[LessEqual[x, 8.5e-59]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3500000000000001e62 or 8.49999999999999933e-59 < x

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 89.0%

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

   if -2.3500000000000001e62 < x < 8.49999999999999933e-59

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 87.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+62} \lor \neg \left(x \leq 8.5 \cdot 10^{-59}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-69} \lor \neg \left(x \leq 2.8 \cdot 10^{-82}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.2e-69) (not (<= x 2.8e-82))) (* x (cos y)) (* z (sin y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.2e-69) || !(x <= 2.8e-82)) {
		tmp = x * cos(y);
	} 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) :: tmp
    if ((x <= (-1.2d-69)) .or. (.not. (x <= 2.8d-82))) then
        tmp = x * cos(y)
    else
        tmp = z * sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.2e-69) || !(x <= 2.8e-82)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = z * Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.2e-69) or not (x <= 2.8e-82):
		tmp = x * math.cos(y)
	else:
		tmp = z * math.sin(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.2e-69) || !(x <= 2.8e-82))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(z * sin(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.2e-69) || ~((x <= 2.8e-82)))
		tmp = x * cos(y);
	else
		tmp = z * sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.2e-69], N[Not[LessEqual[x, 2.8e-82]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2000000000000001e-69 or 2.80000000000000024e-82 < x

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 86.5%

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

   if -1.2000000000000001e-69 < x < 2.80000000000000024e-82

   1. Initial program 99.7%

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

      1. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 75.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-69} \lor \neg \left(x \leq 2.8 \cdot 10^{-82}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.12 \lor \neg \left(y \leq 3.05 \cdot 10^{-33}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.12) (not (<= y 3.05e-33)))
   (* 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 <= -0.12) || !(y <= 3.05e-33)) {
		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 <= (-0.12d0)) .or. (.not. (y <= 3.05d-33))) 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 <= -0.12) || !(y <= 3.05e-33)) {
		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 <= -0.12) or not (y <= 3.05e-33):
		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 <= -0.12) || !(y <= 3.05e-33))
		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 <= -0.12) || ~((y <= 3.05e-33)))
		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, -0.12], N[Not[LessEqual[y, 3.05e-33]], $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 < -0.12 or 3.0500000000000001e-33 < y

   1. Initial program 99.6%

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

      1. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 58.3%

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

   if -0.12 < y < 3.0500000000000001e-33

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.12 \lor \neg \left(y \leq 3.05 \cdot 10^{-33}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.5% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-225}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-59}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.8e-225) x (if (<= x 1.85e-59) (* y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.8e-225) {
		tmp = x;
	} else if (x <= 1.85e-59) {
		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 <= (-8.8d-225)) then
        tmp = x
    else if (x <= 1.85d-59) 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 <= -8.8e-225) {
		tmp = x;
	} else if (x <= 1.85e-59) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8.8e-225:
		tmp = x
	elif x <= 1.85e-59:
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8.8e-225)
		tmp = x;
	elseif (x <= 1.85e-59)
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8.8e-225)
		tmp = x;
	elseif (x <= 1.85e-59)
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8.8e-225], x, If[LessEqual[x, 1.85e-59], N[(y * z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -8.8e-225 or 1.85e-59 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 68.4%

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

      1. flip-+ 33.5%

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

      2. clear-num 33.4%

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

      3. pow2 33.4%

         \[\leadsto \frac{1}{\frac{x - z \cdot \sin y}{\color{blue}{{x}^{2}} - \left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)}} \]

      4. pow2 33.4%

         \[\leadsto \frac{1}{\frac{x - z \cdot \sin y}{{x}^{2} - \color{blue}{{\left(z \cdot \sin y\right)}^{2}}}} \]

   5. Applied egg-rr 33.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - z \cdot \sin y}{{x}^{2} - {\left(z \cdot \sin y\right)}^{2}}}} \]

   6. Taylor expanded in x around inf 51.6%

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

      1. remove-double-div 51.7%

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

      2. *-rgt-identity 51.7%

         \[\leadsto \color{blue}{x \cdot 1} \]

   8. Applied egg-rr 51.7%

      \[\leadsto \color{blue}{x \cdot 1} \]

   if -8.8e-225 < x < 1.85e-59

   1. Initial program 99.7%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 41.4%

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

   6. Taylor expanded in x around 0 29.1%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-225}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-59}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.9% 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. Step-by-step derivation

   1. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in y around 0 51.9%

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

Alternative 8: 4.8% 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 73.8%

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

   1. flip-+ 38.8%

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

   2. clear-num 38.7%

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

   3. pow2 38.7%

      \[\leadsto \frac{1}{\frac{x - z \cdot \sin y}{\color{blue}{{x}^{2}} - \left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)}} \]

   4. pow2 38.7%

      \[\leadsto \frac{1}{\frac{x - z \cdot \sin y}{{x}^{2} - \color{blue}{{\left(z \cdot \sin y\right)}^{2}}}} \]

5. Applied egg-rr 38.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{x - z \cdot \sin y}{{x}^{2} - {\left(z \cdot \sin y\right)}^{2}}}} \]

6. Taylor expanded in x around inf 42.3%

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

   1. remove-double-div 42.4%

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

   2. add-sqr-sqrt 19.0%

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

   3. sqrt-unprod 10.3%

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

   4. sqr-neg 10.3%

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

   5. mul-1-neg 10.3%

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

   6. mul-1-neg 10.3%

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

   7. sqrt-unprod 2.5%

      \[\leadsto \color{blue}{\sqrt{-1 \cdot x} \cdot \sqrt{-1 \cdot x}} \]

   8. add-sqr-sqrt 4.9%

      \[\leadsto \color{blue}{-1 \cdot x} \]

   9. mul-1-neg 4.9%

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

   10. neg-sub0 4.9%

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

8. Applied egg-rr 4.9%

   \[\leadsto \color{blue}{0 - x} \]
9. Step-by-step derivation

   1. neg-sub0 4.9%

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

10. Simplified 4.9%

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

Alternative 9: 38.0% 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 99.8%

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

3. Taylor expanded in y around 0 73.8%

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

   1. flip-+ 38.8%

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

   2. clear-num 38.7%

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

   3. pow2 38.7%

      \[\leadsto \frac{1}{\frac{x - z \cdot \sin y}{\color{blue}{{x}^{2}} - \left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)}} \]

   4. pow2 38.7%

      \[\leadsto \frac{1}{\frac{x - z \cdot \sin y}{{x}^{2} - \color{blue}{{\left(z \cdot \sin y\right)}^{2}}}} \]

5. Applied egg-rr 38.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{x - z \cdot \sin y}{{x}^{2} - {\left(z \cdot \sin y\right)}^{2}}}} \]

6. Taylor expanded in x around inf 42.3%

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

   1. remove-double-div 42.4%

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

   2. *-rgt-identity 42.4%

      \[\leadsto \color{blue}{x \cdot 1} \]

8. Applied egg-rr 42.4%

   \[\leadsto \color{blue}{x \cdot 1} \]

9. Final simplification 42.4%

   \[\leadsto x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))