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

Percentage Accurate: 99.8% → 99.8%

Time: 7.9s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * sin(y)) + (z * cos(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * sin(y)) + (z * cos(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * sin(y)) + (z * cos(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 2: 85.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+102} \lor \neg \left(z \leq -7.8 \cdot 10^{+62} \lor \neg \left(z \leq -0.44\right) \land z \leq 3.9 \cdot 10^{+83}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.9e+102)
         (not (or (<= z -7.8e+62) (and (not (<= z -0.44)) (<= z 3.9e+83)))))
   (* z (cos y))
   (+ (* x (sin y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.9e+102) || !((z <= -7.8e+62) || (!(z <= -0.44) && (z <= 3.9e+83)))) {
		tmp = z * 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 ((z <= (-1.9d+102)) .or. (.not. (z <= (-7.8d+62)) .or. (.not. (z <= (-0.44d0))) .and. (z <= 3.9d+83))) then
        tmp = z * 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 ((z <= -1.9e+102) || !((z <= -7.8e+62) || (!(z <= -0.44) && (z <= 3.9e+83)))) {
		tmp = z * Math.cos(y);
	} else {
		tmp = (x * Math.sin(y)) + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.9e+102) or not ((z <= -7.8e+62) or (not (z <= -0.44) and (z <= 3.9e+83))):
		tmp = z * math.cos(y)
	else:
		tmp = (x * math.sin(y)) + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.9e+102) || !((z <= -7.8e+62) || (!(z <= -0.44) && (z <= 3.9e+83))))
		tmp = Float64(z * cos(y));
	else
		tmp = Float64(Float64(x * sin(y)) + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.9e+102) || ~(((z <= -7.8e+62) || (~((z <= -0.44)) && (z <= 3.9e+83)))))
		tmp = z * cos(y);
	else
		tmp = (x * sin(y)) + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.9e+102], N[Not[Or[LessEqual[z, -7.8e+62], And[N[Not[LessEqual[z, -0.44]], $MachinePrecision], LessEqual[z, 3.9e+83]]]], $MachinePrecision]], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.89999999999999989e102 or -7.8e62 < z < -0.440000000000000002 or 3.9000000000000002e83 < z

   1. Initial program 99.8%

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

      1. add-cube-cbrt 99.7%

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

      2. associate-*l* 99.7%

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

      3. fma-def 99.7%

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

      4. pow2 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 92.4%

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

   if -1.89999999999999989e102 < z < -7.8e62 or -0.440000000000000002 < z < 3.9000000000000002e83

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 89.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+102} \lor \neg \left(z \leq -7.8 \cdot 10^{+62} \lor \neg \left(z \leq -0.44\right) \land z \leq 3.9 \cdot 10^{+83}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y + z\\ \end{array} \]

Alternative 3: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;y \leq -0.019:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-13}:\\ \;\;\;\;z + x \cdot y\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+87} \lor \neg \left(y \leq 2.05 \cdot 10^{+114}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= y -0.019)
     t_0
     (if (<= y 8.8e-13)
       (+ z (* x y))
       (if (or (<= y 1.15e+87) (not (<= y 2.05e+114))) t_0 (* x (sin y)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (y <= -0.019) {
		tmp = t_0;
	} else if (y <= 8.8e-13) {
		tmp = z + (x * y);
	} else if ((y <= 1.15e+87) || !(y <= 2.05e+114)) {
		tmp = t_0;
	} else {
		tmp = x * 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 = z * cos(y)
    if (y <= (-0.019d0)) then
        tmp = t_0
    else if (y <= 8.8d-13) then
        tmp = z + (x * y)
    else if ((y <= 1.15d+87) .or. (.not. (y <= 2.05d+114))) then
        tmp = t_0
    else
        tmp = x * sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.cos(y);
	double tmp;
	if (y <= -0.019) {
		tmp = t_0;
	} else if (y <= 8.8e-13) {
		tmp = z + (x * y);
	} else if ((y <= 1.15e+87) || !(y <= 2.05e+114)) {
		tmp = t_0;
	} else {
		tmp = x * Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if y <= -0.019:
		tmp = t_0
	elif y <= 8.8e-13:
		tmp = z + (x * y)
	elif (y <= 1.15e+87) or not (y <= 2.05e+114):
		tmp = t_0
	else:
		tmp = x * math.sin(y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -0.019)
		tmp = t_0;
	elseif (y <= 8.8e-13)
		tmp = Float64(z + Float64(x * y));
	elseif ((y <= 1.15e+87) || !(y <= 2.05e+114))
		tmp = t_0;
	else
		tmp = Float64(x * sin(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	tmp = 0.0;
	if (y <= -0.019)
		tmp = t_0;
	elseif (y <= 8.8e-13)
		tmp = z + (x * y);
	elseif ((y <= 1.15e+87) || ~((y <= 2.05e+114)))
		tmp = t_0;
	else
		tmp = x * sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.019], t$95$0, If[LessEqual[y, 8.8e-13], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 1.15e+87], N[Not[LessEqual[y, 2.05e+114]], $MachinePrecision]], t$95$0, N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0189999999999999995 or 8.79999999999999986e-13 < y < 1.1500000000000001e87 or 2.05e114 < y

   1. Initial program 99.6%

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

      1. add-cube-cbrt 98.9%

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

      2. associate-*l* 98.9%

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

      3. fma-def 98.9%

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

      4. pow2 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in x around 0 61.1%

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

   if -0.0189999999999999995 < y < 8.79999999999999986e-13

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.3%

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

   if 1.1500000000000001e87 < y < 2.05e114

   1. Initial program 99.7%

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

      1. add-cube-cbrt 98.4%

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

      2. associate-*l* 98.4%

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

      3. fma-def 98.4%

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

      4. pow2 98.4%

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

   3. Applied egg-rr 98.4%

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

   4. Taylor expanded in z around 0 99.7%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\sin y \cdot x\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 99.7%

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

      2. *-lft-identity 99.7%

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

      3. *-commutative 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.019:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-13}:\\ \;\;\;\;z + x \cdot y\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+87} \lor \neg \left(y \leq 2.05 \cdot 10^{+114}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \]

Alternative 4: 74.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0105 \lor \neg \left(y \leq 0.0037\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0105) (not (<= y 0.0037))) (* x (sin y)) (+ z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0105) || !(y <= 0.0037)) {
		tmp = x * sin(y);
	} else {
		tmp = z + (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 <= (-0.0105d0)) .or. (.not. (y <= 0.0037d0))) then
        tmp = x * sin(y)
    else
        tmp = z + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0105) || !(y <= 0.0037)) {
		tmp = x * Math.sin(y);
	} else {
		tmp = z + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0105) or not (y <= 0.0037):
		tmp = x * math.sin(y)
	else:
		tmp = z + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0105) || !(y <= 0.0037))
		tmp = Float64(x * sin(y));
	else
		tmp = Float64(z + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.0105) || ~((y <= 0.0037)))
		tmp = x * sin(y);
	else
		tmp = z + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0105], N[Not[LessEqual[y, 0.0037]], $MachinePrecision]], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0105000000000000007 or 0.0037000000000000002 < y

   1. Initial program 99.6%

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

      1. add-cube-cbrt 98.9%

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

      2. associate-*l* 98.9%

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

      3. fma-def 98.9%

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

      4. pow2 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in z around 0 44.3%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\sin y \cdot x\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 44.3%

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

      2. *-lft-identity 44.3%

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

      3. *-commutative 44.3%

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

   6. Simplified 44.3%

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

   if -0.0105000000000000007 < y < 0.0037000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.6%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0105 \lor \neg \left(y \leq 0.0037\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]

Alternative 5: 41.9% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-102}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-226}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-68}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-49}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.1e-102)
   z
   (if (<= z 3e-226)
     (* x y)
     (if (<= z 8e-68) z (if (<= z 6.6e-49) (* x y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e-102) {
		tmp = z;
	} else if (z <= 3e-226) {
		tmp = x * y;
	} else if (z <= 8e-68) {
		tmp = z;
	} else if (z <= 6.6e-49) {
		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 (z <= (-2.1d-102)) then
        tmp = z
    else if (z <= 3d-226) then
        tmp = x * y
    else if (z <= 8d-68) then
        tmp = z
    else if (z <= 6.6d-49) 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 (z <= -2.1e-102) {
		tmp = z;
	} else if (z <= 3e-226) {
		tmp = x * y;
	} else if (z <= 8e-68) {
		tmp = z;
	} else if (z <= 6.6e-49) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.1e-102:
		tmp = z
	elif z <= 3e-226:
		tmp = x * y
	elif z <= 8e-68:
		tmp = z
	elif z <= 6.6e-49:
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.1e-102)
		tmp = z;
	elseif (z <= 3e-226)
		tmp = Float64(x * y);
	elseif (z <= 8e-68)
		tmp = z;
	elseif (z <= 6.6e-49)
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.1e-102)
		tmp = z;
	elseif (z <= 3e-226)
		tmp = x * y;
	elseif (z <= 8e-68)
		tmp = z;
	elseif (z <= 6.6e-49)
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.1e-102], z, If[LessEqual[z, 3e-226], N[(x * y), $MachinePrecision], If[LessEqual[z, 8e-68], z, If[LessEqual[z, 6.6e-49], N[(x * y), $MachinePrecision], z]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1e-102 or 2.99999999999999995e-226 < z < 8.00000000000000053e-68 or 6.6e-49 < z

   1. Initial program 99.8%

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

      1. add-cube-cbrt 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. pow2 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in y around 0 49.5%

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

   if -2.1e-102 < z < 2.99999999999999995e-226 or 8.00000000000000053e-68 < z < 6.6e-49

   1. Initial program 99.8%

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

      1. add-cube-cbrt 98.1%

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

      2. associate-*l* 98.1%

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

      3. fma-def 98.1%

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

      4. pow2 98.1%

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

   3. Applied egg-rr 98.1%

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

   4. Taylor expanded in z around 0 85.2%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\sin y \cdot x\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 85.2%

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

      2. *-lft-identity 85.2%

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

      3. *-commutative 85.2%

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

   6. Simplified 85.2%

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

   7. Taylor expanded in y around 0 44.8%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-102}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-226}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-68}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-49}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 6: 52.0% 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. Taylor expanded in y around 0 56.2%

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

3. Final simplification 56.2%

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

Alternative 7: 39.7% 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. Step-by-step derivation

   1. add-cube-cbrt 99.2%

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

   2. associate-*l* 99.1%

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

   3. fma-def 99.1%

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

   4. pow2 99.1%

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

3. Applied egg-rr 99.1%

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

4. Taylor expanded in y around 0 40.3%

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

5. Final simplification 40.3%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023244 
(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))))