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

Percentage Accurate: 99.8% → 99.8%

Time: 7.2s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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 2: 75.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.00062:\\ \;\;\;\;y \cdot z + \left(x + -0.5 \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\\ \mathbf{elif}\;y \leq 3.45 \cdot 10^{+47} \lor \neg \left(y \leq 1.6 \cdot 10^{+83}\right) \land y \leq 8.4 \cdot 10^{+245}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (<= y -6.2e-6)
     t_0
     (if (<= y 0.00062)
       (+ (* y z) (+ x (* -0.5 (* y (* x y)))))
       (if (or (<= y 3.45e+47) (and (not (<= y 1.6e+83)) (<= y 8.4e+245)))
         t_0
         (* x (cos y)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double tmp;
	if (y <= -6.2e-6) {
		tmp = t_0;
	} else if (y <= 0.00062) {
		tmp = (y * z) + (x + (-0.5 * (y * (x * y))));
	} else if ((y <= 3.45e+47) || (!(y <= 1.6e+83) && (y <= 8.4e+245))) {
		tmp = t_0;
	} else {
		tmp = x * cos(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 * sin(y)
    if (y <= (-6.2d-6)) then
        tmp = t_0
    else if (y <= 0.00062d0) then
        tmp = (y * z) + (x + ((-0.5d0) * (y * (x * y))))
    else if ((y <= 3.45d+47) .or. (.not. (y <= 1.6d+83)) .and. (y <= 8.4d+245)) then
        tmp = t_0
    else
        tmp = x * cos(y)
    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 tmp;
	if (y <= -6.2e-6) {
		tmp = t_0;
	} else if (y <= 0.00062) {
		tmp = (y * z) + (x + (-0.5 * (y * (x * y))));
	} else if ((y <= 3.45e+47) || (!(y <= 1.6e+83) && (y <= 8.4e+245))) {
		tmp = t_0;
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	tmp = 0
	if y <= -6.2e-6:
		tmp = t_0
	elif y <= 0.00062:
		tmp = (y * z) + (x + (-0.5 * (y * (x * y))))
	elif (y <= 3.45e+47) or (not (y <= 1.6e+83) and (y <= 8.4e+245)):
		tmp = t_0
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	tmp = 0.0
	if (y <= -6.2e-6)
		tmp = t_0;
	elseif (y <= 0.00062)
		tmp = Float64(Float64(y * z) + Float64(x + Float64(-0.5 * Float64(y * Float64(x * y)))));
	elseif ((y <= 3.45e+47) || (!(y <= 1.6e+83) && (y <= 8.4e+245)))
		tmp = t_0;
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	tmp = 0.0;
	if (y <= -6.2e-6)
		tmp = t_0;
	elseif (y <= 0.00062)
		tmp = (y * z) + (x + (-0.5 * (y * (x * y))));
	elseif ((y <= 3.45e+47) || (~((y <= 1.6e+83)) && (y <= 8.4e+245)))
		tmp = t_0;
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.2e-6], t$95$0, If[LessEqual[y, 0.00062], N[(N[(y * z), $MachinePrecision] + N[(x + N[(-0.5 * N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 3.45e+47], And[N[Not[LessEqual[y, 1.6e+83]], $MachinePrecision], LessEqual[y, 8.4e+245]]], t$95$0, N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.1999999999999999e-6 or 6.2e-4 < y < 3.4500000000000002e47 or 1.5999999999999999e83 < y < 8.39999999999999984e245

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 68.7%

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

   if -6.1999999999999999e-6 < y < 6.2e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.6%

      \[\leadsto \color{blue}{y \cdot z + \left(-0.5 \cdot \left({y}^{2} \cdot x\right) + x\right)} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 91.4%

         \[\leadsto y \cdot z + \left(-0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2} \cdot x\right)\right)} + x\right) \]

      2. expm1-udef 91.4%

         \[\leadsto y \cdot z + \left(-0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({y}^{2} \cdot x\right)} - 1\right)} + x\right) \]

      3. unpow2 91.4%

         \[\leadsto y \cdot z + \left(-0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(y \cdot y\right)} \cdot x\right)} - 1\right) + x\right) \]

      4. associate-*l* 91.4%

         \[\leadsto y \cdot z + \left(-0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left(y \cdot x\right)}\right)} - 1\right) + x\right) \]

   4. Applied egg-rr 91.4%

      \[\leadsto y \cdot z + \left(-0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left(y \cdot x\right)\right)} - 1\right)} + x\right) \]
   5. Step-by-step derivation

      1. expm1-def 91.4%

         \[\leadsto y \cdot z + \left(-0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(y \cdot x\right)\right)\right)} + x\right) \]

      2. expm1-log1p 99.6%

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

   6. Simplified 99.6%

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

   if 3.4500000000000002e47 < y < 1.5999999999999999e83 or 8.39999999999999984e245 < y

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. add-cube-cbrt 98.2%

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

      3. associate-*r* 98.2%

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

      4. fma-def 98.2%

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

      5. pow2 98.2%

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

   3. Applied egg-rr 98.2%

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

   4. Taylor expanded in z around 0 81.3%

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

      1. pow-base-1 81.3%

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

      2. *-lft-identity 81.3%

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

   6. Simplified 81.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq 0.00062:\\ \;\;\;\;y \cdot z + \left(x + -0.5 \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\\ \mathbf{elif}\;y \leq 3.45 \cdot 10^{+47} \lor \neg \left(y \leq 1.6 \cdot 10^{+83}\right) \land y \leq 8.4 \cdot 10^{+245}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 3: 84.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+223} \lor \neg \left(x \leq 6.5 \cdot 10^{+112}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.8e+223) (not (<= x 6.5e+112)))
   (* x (cos y))
   (+ x (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.8e+223) || !(x <= 6.5e+112)) {
		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 <= (-5.8d+223)) .or. (.not. (x <= 6.5d+112))) 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 <= -5.8e+223) || !(x <= 6.5e+112)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.8e+223) or not (x <= 6.5e+112):
		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 <= -5.8e+223) || !(x <= 6.5e+112))
		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 <= -5.8e+223) || ~((x <= 6.5e+112)))
		tmp = x * cos(y);
	else
		tmp = x + (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.8e+223], N[Not[LessEqual[x, 6.5e+112]], $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 < -5.8000000000000004e223 or 6.4999999999999998e112 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. add-cube-cbrt 98.0%

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

      3. associate-*r* 98.0%

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

      4. fma-def 98.0%

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

      5. pow2 98.0%

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

   3. Applied egg-rr 98.0%

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

   4. Taylor expanded in z around 0 94.7%

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

      1. pow-base-1 94.7%

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

      2. *-lft-identity 94.7%

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

   6. Simplified 94.7%

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

   if -5.8000000000000004e223 < x < 6.4999999999999998e112

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 86.1%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+223} \lor \neg \left(x \leq 6.5 \cdot 10^{+112}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \]

Alternative 4: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-6} \lor \neg \left(y \leq 0.00062\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z + \left(x + -0.5 \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.2e-6) (not (<= y 0.00062)))
   (* z (sin y))
   (+ (* y z) (+ x (* -0.5 (* y (* x y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.2e-6) || !(y <= 0.00062)) {
		tmp = z * sin(y);
	} else {
		tmp = (y * z) + (x + (-0.5 * (y * (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 <= (-6.2d-6)) .or. (.not. (y <= 0.00062d0))) then
        tmp = z * sin(y)
    else
        tmp = (y * z) + (x + ((-0.5d0) * (y * (x * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.2e-6) || !(y <= 0.00062)) {
		tmp = z * Math.sin(y);
	} else {
		tmp = (y * z) + (x + (-0.5 * (y * (x * y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6.2e-6) or not (y <= 0.00062):
		tmp = z * math.sin(y)
	else:
		tmp = (y * z) + (x + (-0.5 * (y * (x * y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.2e-6) || !(y <= 0.00062))
		tmp = Float64(z * sin(y));
	else
		tmp = Float64(Float64(y * z) + Float64(x + Float64(-0.5 * Float64(y * Float64(x * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -6.2e-6) || ~((y <= 0.00062)))
		tmp = z * sin(y);
	else
		tmp = (y * z) + (x + (-0.5 * (y * (x * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.2e-6], N[Not[LessEqual[y, 0.00062]], $MachinePrecision]], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * z), $MachinePrecision] + N[(x + N[(-0.5 * N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.1999999999999999e-6 or 6.2e-4 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 58.7%

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

   if -6.1999999999999999e-6 < y < 6.2e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.6%

      \[\leadsto \color{blue}{y \cdot z + \left(-0.5 \cdot \left({y}^{2} \cdot x\right) + x\right)} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 91.4%

         \[\leadsto y \cdot z + \left(-0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2} \cdot x\right)\right)} + x\right) \]

      2. expm1-udef 91.4%

         \[\leadsto y \cdot z + \left(-0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({y}^{2} \cdot x\right)} - 1\right)} + x\right) \]

      3. unpow2 91.4%

         \[\leadsto y \cdot z + \left(-0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(y \cdot y\right)} \cdot x\right)} - 1\right) + x\right) \]

      4. associate-*l* 91.4%

         \[\leadsto y \cdot z + \left(-0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left(y \cdot x\right)}\right)} - 1\right) + x\right) \]

   4. Applied egg-rr 91.4%

      \[\leadsto y \cdot z + \left(-0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left(y \cdot x\right)\right)} - 1\right)} + x\right) \]
   5. Step-by-step derivation

      1. expm1-def 91.4%

         \[\leadsto y \cdot z + \left(-0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(y \cdot x\right)\right)\right)} + x\right) \]

      2. expm1-log1p 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-6} \lor \neg \left(y \leq 0.00062\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z + \left(x + -0.5 \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\\ \end{array} \]

Alternative 5: 41.8% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+94}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+93}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+303}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3e+94)
   (* y z)
   (if (<= z 1.9e+93) x (if (<= z 1.6e+303) (* y z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e+94) {
		tmp = y * z;
	} else if (z <= 1.9e+93) {
		tmp = x;
	} else if (z <= 1.6e+303) {
		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 (z <= (-3d+94)) then
        tmp = y * z
    else if (z <= 1.9d+93) then
        tmp = x
    else if (z <= 1.6d+303) 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 (z <= -3e+94) {
		tmp = y * z;
	} else if (z <= 1.9e+93) {
		tmp = x;
	} else if (z <= 1.6e+303) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3e+94:
		tmp = y * z
	elif z <= 1.9e+93:
		tmp = x
	elif z <= 1.6e+303:
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3e+94)
		tmp = Float64(y * z);
	elseif (z <= 1.9e+93)
		tmp = x;
	elseif (z <= 1.6e+303)
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3e+94)
		tmp = y * z;
	elseif (z <= 1.9e+93)
		tmp = x;
	elseif (z <= 1.6e+303)
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3e+94], N[(y * z), $MachinePrecision], If[LessEqual[z, 1.9e+93], x, If[LessEqual[z, 1.6e+303], N[(y * z), $MachinePrecision], x]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.0000000000000001e94 or 1.8999999999999999e93 < z < 1.6000000000000001e303

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 49.4%

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

   3. Taylor expanded in z around inf 35.2%

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

   if -3.0000000000000001e94 < z < 1.8999999999999999e93 or 1.6000000000000001e303 < z

   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. add-cube-cbrt 99.4%

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

      3. fma-def 99.4%

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

      4. pow2 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in y around 0 51.3%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+94}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+93}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+303}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 53.9% 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. Taylor expanded in y around 0 55.3%

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

3. Final simplification 55.3%

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

Alternative 7: 39.2% 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. add-cube-cbrt 99.1%

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

   3. fma-def 99.1%

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

   4. pow2 99.1%

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

3. Applied egg-rr 99.1%

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

4. Taylor expanded in y around 0 41.3%

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

5. Final simplification 41.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023185 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))