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

Percentage Accurate: 99.8% → 99.8%

Time: 7.6s

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+214}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.00072:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.0155:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+114} \lor \neg \left(y \leq 6.8 \cdot 10^{+227}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (* x (cos y))))
   (if (<= y -4.5e+214)
     t_0
     (if (<= y -0.00072)
       t_1
       (if (<= y 0.0155)
         (+ x (* y z))
         (if (or (<= y 7e+114) (not (<= y 6.8e+227))) t_1 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 <= -4.5e+214) {
		tmp = t_0;
	} else if (y <= -0.00072) {
		tmp = t_1;
	} else if (y <= 0.0155) {
		tmp = x + (y * z);
	} else if ((y <= 7e+114) || !(y <= 6.8e+227)) {
		tmp = t_1;
	} 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 <= (-4.5d+214)) then
        tmp = t_0
    else if (y <= (-0.00072d0)) then
        tmp = t_1
    else if (y <= 0.0155d0) then
        tmp = x + (y * z)
    else if ((y <= 7d+114) .or. (.not. (y <= 6.8d+227))) then
        tmp = t_1
    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 <= -4.5e+214) {
		tmp = t_0;
	} else if (y <= -0.00072) {
		tmp = t_1;
	} else if (y <= 0.0155) {
		tmp = x + (y * z);
	} else if ((y <= 7e+114) || !(y <= 6.8e+227)) {
		tmp = t_1;
	} 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 <= -4.5e+214:
		tmp = t_0
	elif y <= -0.00072:
		tmp = t_1
	elif y <= 0.0155:
		tmp = x + (y * z)
	elif (y <= 7e+114) or not (y <= 6.8e+227):
		tmp = t_1
	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 <= -4.5e+214)
		tmp = t_0;
	elseif (y <= -0.00072)
		tmp = t_1;
	elseif (y <= 0.0155)
		tmp = Float64(x + Float64(y * z));
	elseif ((y <= 7e+114) || !(y <= 6.8e+227))
		tmp = t_1;
	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 <= -4.5e+214)
		tmp = t_0;
	elseif (y <= -0.00072)
		tmp = t_1;
	elseif (y <= 0.0155)
		tmp = x + (y * z);
	elseif ((y <= 7e+114) || ~((y <= 6.8e+227)))
		tmp = t_1;
	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, -4.5e+214], t$95$0, If[LessEqual[y, -0.00072], t$95$1, If[LessEqual[y, 0.0155], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 7e+114], N[Not[LessEqual[y, 6.8e+227]], $MachinePrecision]], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.49999999999999968e214 or 7.0000000000000001e114 < y < 6.79999999999999979e227

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 71.5%

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

   if -4.49999999999999968e214 < y < -7.20000000000000045e-4 or 0.0155 < y < 7.0000000000000001e114 or 6.79999999999999979e227 < y

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. add-cube-cbrt 99.0%

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

      4. associate-*r* 99.0%

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

      5. fma-def 99.0%

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

      6. pow2 99.0%

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

   3. Applied egg-rr 99.0%

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

   4. Taylor expanded in z around 0 65.1%

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

   if -7.20000000000000045e-4 < y < 0.0155

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+214}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -0.00072:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.0155:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+114} \lor \neg \left(y \leq 6.8 \cdot 10^{+227}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \]

Alternative 4: 85.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.021 \lor \neg \left(x \leq 4.7 \cdot 10^{+115}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.021) (not (<= x 4.7e+115)))
   (* x (cos y))
   (+ x (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.021) || !(x <= 4.7e+115)) {
		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 <= (-0.021d0)) .or. (.not. (x <= 4.7d+115))) 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 <= -0.021) || !(x <= 4.7e+115)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -0.021) or not (x <= 4.7e+115):
		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 <= -0.021) || !(x <= 4.7e+115))
		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 <= -0.021) || ~((x <= 4.7e+115)))
		tmp = x * cos(y);
	else
		tmp = x + (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.021], N[Not[LessEqual[x, 4.7e+115]], $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 < -0.0210000000000000013 or 4.6999999999999996e115 < 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 99.6%

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

      4. associate-*r* 99.6%

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

      5. fma-def 99.6%

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

      6. pow2 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in z around 0 92.2%

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

   if -0.0210000000000000013 < x < 4.6999999999999996e115

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 88.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.021 \lor \neg \left(x \leq 4.7 \cdot 10^{+115}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \]

Alternative 5: 73.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -135000000000 \lor \neg \left(y \leq 1.85 \cdot 10^{+28}\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -135000000000.0) (not (<= y 1.85e+28)))
   (* z (sin y))
   (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -135000000000.0) || !(y <= 1.85e+28)) {
		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 <= (-135000000000.0d0)) .or. (.not. (y <= 1.85d+28))) 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 <= -135000000000.0) || !(y <= 1.85e+28)) {
		tmp = z * Math.sin(y);
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -135000000000.0) or not (y <= 1.85e+28):
		tmp = z * math.sin(y)
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -135000000000.0) || !(y <= 1.85e+28))
		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 <= -135000000000.0) || ~((y <= 1.85e+28)))
		tmp = z * sin(y);
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -135000000000.0], N[Not[LessEqual[y, 1.85e+28]], $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 < -1.35e11 or 1.85e28 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 48.9%

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

   if -1.35e11 < y < 1.85e28

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 96.8%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -135000000000 \lor \neg \left(y \leq 1.85 \cdot 10^{+28}\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 6: 40.6% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-132}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-294}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.4e-132) x (if (<= x -8.2e-294) (* y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.4e-132) {
		tmp = x;
	} else if (x <= -8.2e-294) {
		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 <= (-3.4d-132)) then
        tmp = x
    else if (x <= (-8.2d-294)) 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 <= -3.4e-132) {
		tmp = x;
	} else if (x <= -8.2e-294) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.4e-132:
		tmp = x
	elif x <= -8.2e-294:
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.4e-132)
		tmp = x;
	elseif (x <= -8.2e-294)
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.4e-132)
		tmp = x;
	elseif (x <= -8.2e-294)
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.4e-132], x, If[LessEqual[x, -8.2e-294], N[(y * z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.39999999999999983e-132 or -8.1999999999999998e-294 < 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 99.2%

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

      4. associate-*r* 99.2%

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

      5. fma-def 99.2%

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

      6. pow2 99.2%

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

   3. Applied egg-rr 99.2%

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

   4. Taylor expanded in y around 0 45.6%

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

   if -3.39999999999999983e-132 < x < -8.1999999999999998e-294

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 69.1%

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

   3. Taylor expanded in z around inf 51.7%

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

      1. *-commutative 51.7%

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

   5. Simplified 51.7%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-132}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-294}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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 57.2%

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

3. Final simplification 57.2%

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

Alternative 8: 39.1% 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%

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

   4. associate-*r* 99.1%

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

   5. fma-def 99.1%

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

   6. pow2 99.1%

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

3. Applied egg-rr 99.1%

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

4. Taylor expanded in y around 0 42.6%

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

5. Final simplification 42.6%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023263 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))