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

Percentage Accurate: 99.8% → 99.8%

Time: 8.9s

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, 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-def 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. Final simplification 99.8%

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

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. Final simplification 99.8%

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

Alternative 3: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+206}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.036:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (* x (cos y))))
   (if (<= y -3.2e+206)
     t_0
     (if (<= y -3e+162)
       t_1
       (if (<= y -1.35e+55)
         t_0
         (if (<= y -0.036) t_1 (if (<= y 2.4e-7) (+ x (* y z)) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = x * cos(y);
	double tmp;
	if (y <= -3.2e+206) {
		tmp = t_0;
	} else if (y <= -3e+162) {
		tmp = t_1;
	} else if (y <= -1.35e+55) {
		tmp = t_0;
	} else if (y <= -0.036) {
		tmp = t_1;
	} else if (y <= 2.4e-7) {
		tmp = x + (y * z);
	} 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) :: t_1
    real(8) :: tmp
    t_0 = z * sin(y)
    t_1 = x * cos(y)
    if (y <= (-3.2d+206)) then
        tmp = t_0
    else if (y <= (-3d+162)) then
        tmp = t_1
    else if (y <= (-1.35d+55)) then
        tmp = t_0
    else if (y <= (-0.036d0)) then
        tmp = t_1
    else if (y <= 2.4d-7) then
        tmp = x + (y * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.sin(y);
	double t_1 = x * Math.cos(y);
	double tmp;
	if (y <= -3.2e+206) {
		tmp = t_0;
	} else if (y <= -3e+162) {
		tmp = t_1;
	} else if (y <= -1.35e+55) {
		tmp = t_0;
	} else if (y <= -0.036) {
		tmp = t_1;
	} else if (y <= 2.4e-7) {
		tmp = x + (y * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	t_1 = x * math.cos(y)
	tmp = 0
	if y <= -3.2e+206:
		tmp = t_0
	elif y <= -3e+162:
		tmp = t_1
	elif y <= -1.35e+55:
		tmp = t_0
	elif y <= -0.036:
		tmp = t_1
	elif y <= 2.4e-7:
		tmp = x + (y * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -3.2e+206)
		tmp = t_0;
	elseif (y <= -3e+162)
		tmp = t_1;
	elseif (y <= -1.35e+55)
		tmp = t_0;
	elseif (y <= -0.036)
		tmp = t_1;
	elseif (y <= 2.4e-7)
		tmp = Float64(x + Float64(y * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	t_1 = x * cos(y);
	tmp = 0.0;
	if (y <= -3.2e+206)
		tmp = t_0;
	elseif (y <= -3e+162)
		tmp = t_1;
	elseif (y <= -1.35e+55)
		tmp = t_0;
	elseif (y <= -0.036)
		tmp = t_1;
	elseif (y <= 2.4e-7)
		tmp = x + (y * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e+206], t$95$0, If[LessEqual[y, -3e+162], t$95$1, If[LessEqual[y, -1.35e+55], t$95$0, If[LessEqual[y, -0.036], t$95$1, If[LessEqual[y, 2.4e-7], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.20000000000000005e206 or -2.9999999999999998e162 < y < -1.34999999999999988e55 or 2.39999999999999979e-7 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 64.0%

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

   if -3.20000000000000005e206 < y < -2.9999999999999998e162 or -1.34999999999999988e55 < y < -0.0359999999999999973

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. *-commutative 99.4%

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

      3. add-cube-cbrt 99.2%

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

      4. associate-*l* 99.2%

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

      5. fma-def 99.2%

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

      6. pow2 99.2%

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

   3. Applied egg-rr 99.2%

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

   4. Taylor expanded in z around 0 80.2%

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

   if -0.0359999999999999973 < y < 2.39999999999999979e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+206}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+162}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{+55}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -0.036:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \]

Alternative 4: 86.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+35} \lor \neg \left(x \leq 3.25 \cdot 10^{+62}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.1e+35) (not (<= x 3.25e+62)))
   (* x (cos y))
   (+ x (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.1e+35) || !(x <= 3.25e+62)) {
		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 <= (-2.1d+35)) .or. (.not. (x <= 3.25d+62))) 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 <= -2.1e+35) || !(x <= 3.25e+62)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.1e+35) or not (x <= 3.25e+62):
		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 <= -2.1e+35) || !(x <= 3.25e+62))
		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 <= -2.1e+35) || ~((x <= 3.25e+62)))
		tmp = x * cos(y);
	else
		tmp = x + (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.1e+35], N[Not[LessEqual[x, 3.25e+62]], $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 < -2.0999999999999999e35 or 3.2500000000000001e62 < x

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. add-cube-cbrt 99.7%

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

      4. associate-*l* 99.6%

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

      5. fma-def 99.6%

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

      6. pow2 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in z around 0 89.1%

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

   if -2.0999999999999999e35 < x < 3.2500000000000001e62

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 91.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+35} \lor \neg \left(x \leq 3.25 \cdot 10^{+62}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \]

Alternative 5: 73.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+19} \lor \neg \left(y \leq 2.4 \cdot 10^{-7}\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.4e+19) (not (<= y 2.4e-7))) (* z (sin y)) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.4e+19) || !(y <= 2.4e-7)) {
		tmp = z * sin(y);
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-5.4d+19)) .or. (.not. (y <= 2.4d-7))) then
        tmp = z * sin(y)
    else
        tmp = x + (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.4e+19) || !(y <= 2.4e-7)) {
		tmp = z * Math.sin(y);
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.4e+19) or not (y <= 2.4e-7):
		tmp = z * math.sin(y)
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.4e+19) || !(y <= 2.4e-7))
		tmp = Float64(z * sin(y));
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5.4e+19) || ~((y <= 2.4e-7)))
		tmp = z * sin(y);
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.4e+19], N[Not[LessEqual[y, 2.4e-7]], $MachinePrecision]], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4e19 or 2.39999999999999979e-7 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 58.7%

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

   if -5.4e19 < y < 2.39999999999999979e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+19} \lor \neg \left(y \leq 2.4 \cdot 10^{-7}\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 6: 41.7% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-203}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-148}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.2e-203) x (if (<= x 1.15e-148) (* y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.2e-203) {
		tmp = x;
	} else if (x <= 1.15e-148) {
		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 <= (-3.2d-203)) then
        tmp = x
    else if (x <= 1.15d-148) 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 <= -3.2e-203) {
		tmp = x;
	} else if (x <= 1.15e-148) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.2e-203:
		tmp = x
	elif x <= 1.15e-148:
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.2e-203)
		tmp = x;
	elseif (x <= 1.15e-148)
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.2e-203)
		tmp = x;
	elseif (x <= 1.15e-148)
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.2e-203], x, If[LessEqual[x, 1.15e-148], N[(y * z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2e-203 or 1.14999999999999999e-148 < x

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. add-cube-cbrt 99.3%

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

      4. associate-*l* 99.3%

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

      5. fma-def 99.3%

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

      6. pow2 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in y around 0 47.3%

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

   if -3.2e-203 < x < 1.14999999999999999e-148

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0 46.1%

      \[\leadsto \color{blue}{y \cdot z + \left(-0.5 \cdot \left({y}^{2} \cdot x\right) + x\right)} \]

   3. Taylor expanded in z around inf 41.1%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-203}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-148}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 52.4% 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. Taylor expanded in y around 0 54.3%

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

3. Final simplification 54.3%

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

Alternative 8: 39.0% 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. Step-by-step derivation

   1. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. add-cube-cbrt 99.1%

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

   4. associate-*l* 99.1%

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

   5. fma-def 99.1%

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

   6. pow2 99.1%

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

3. Applied egg-rr 99.1%

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

4. Taylor expanded in y around 0 39.5%

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

5. Final simplification 39.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023274 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))