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

Percentage Accurate: 99.8% → 99.8%

Time: 8.0s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

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 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -8 \cdot 10^{+242}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.18:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 30500:\\ \;\;\;\;z \cdot y + \left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{+209} \lor \neg \left(y \leq 3.6 \cdot 10^{+233}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (* x (cos y))))
   (if (<= y -8e+242)
     t_0
     (if (<= y -0.18)
       t_1
       (if (<= y 30500.0)
         (+ (* z y) (+ x (* -0.5 (* x (* y y)))))
         (if (or (<= y 3.05e+209) (not (<= y 3.6e+233))) 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 <= -8e+242) {
		tmp = t_0;
	} else if (y <= -0.18) {
		tmp = t_1;
	} else if (y <= 30500.0) {
		tmp = (z * y) + (x + (-0.5 * (x * (y * y))));
	} else if ((y <= 3.05e+209) || !(y <= 3.6e+233)) {
		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 <= (-8d+242)) then
        tmp = t_0
    else if (y <= (-0.18d0)) then
        tmp = t_1
    else if (y <= 30500.0d0) then
        tmp = (z * y) + (x + ((-0.5d0) * (x * (y * y))))
    else if ((y <= 3.05d+209) .or. (.not. (y <= 3.6d+233))) 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 <= -8e+242) {
		tmp = t_0;
	} else if (y <= -0.18) {
		tmp = t_1;
	} else if (y <= 30500.0) {
		tmp = (z * y) + (x + (-0.5 * (x * (y * y))));
	} else if ((y <= 3.05e+209) || !(y <= 3.6e+233)) {
		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 <= -8e+242:
		tmp = t_0
	elif y <= -0.18:
		tmp = t_1
	elif y <= 30500.0:
		tmp = (z * y) + (x + (-0.5 * (x * (y * y))))
	elif (y <= 3.05e+209) or not (y <= 3.6e+233):
		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 <= -8e+242)
		tmp = t_0;
	elseif (y <= -0.18)
		tmp = t_1;
	elseif (y <= 30500.0)
		tmp = Float64(Float64(z * y) + Float64(x + Float64(-0.5 * Float64(x * Float64(y * y)))));
	elseif ((y <= 3.05e+209) || !(y <= 3.6e+233))
		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 <= -8e+242)
		tmp = t_0;
	elseif (y <= -0.18)
		tmp = t_1;
	elseif (y <= 30500.0)
		tmp = (z * y) + (x + (-0.5 * (x * (y * y))));
	elseif ((y <= 3.05e+209) || ~((y <= 3.6e+233)))
		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, -8e+242], t$95$0, If[LessEqual[y, -0.18], t$95$1, If[LessEqual[y, 30500.0], N[(N[(z * y), $MachinePrecision] + N[(x + N[(-0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 3.05e+209], N[Not[LessEqual[y, 3.6e+233]], $MachinePrecision]], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000041e242 or 3.05000000000000015e209 < y < 3.5999999999999998e233

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 75.1%

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

   if -8.00000000000000041e242 < y < -0.17999999999999999 or 30500 < y < 3.05000000000000015e209 or 3.5999999999999998e233 < y

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. add-cube-cbrt 98.7%

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

      3. associate-*l* 98.6%

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

      4. fma-def 98.6%

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

      5. pow2 98.6%

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

   3. Applied egg-rr 98.6%

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

   4. Taylor expanded in x around inf 63.9%

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

      1. pow-base-1 63.9%

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

      2. *-lft-identity 63.9%

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

   6. Simplified 63.9%

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

   if -0.17999999999999999 < y < 30500

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.7%

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

      1. expm1-log1p-u 90.3%

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

      2. expm1-udef 90.3%

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

      3. *-commutative 90.3%

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

      4. unpow2 90.3%

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

   4. Applied egg-rr 90.3%

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

      1. expm1-def 90.3%

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

      2. expm1-log1p 97.7%

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

      3. unpow2 97.7%

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

      4. *-commutative 97.7%

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

      5. unpow2 97.7%

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

   6. Simplified 97.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+242}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -0.18:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 30500:\\ \;\;\;\;z \cdot y + \left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{+209} \lor \neg \left(y \leq 3.6 \cdot 10^{+233}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \]

Alternative 4: 85.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+140} \lor \neg \left(x \leq 6.8 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.95e+140) (not (<= x 6.8e-12)))
   (* x (cos y))
   (+ x (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.95e+140) || !(x <= 6.8e-12)) {
		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 <= (-1.95d+140)) .or. (.not. (x <= 6.8d-12))) 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 <= -1.95e+140) || !(x <= 6.8e-12)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.95e+140) or not (x <= 6.8e-12):
		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 <= -1.95e+140) || !(x <= 6.8e-12))
		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 <= -1.95e+140) || ~((x <= 6.8e-12)))
		tmp = x * cos(y);
	else
		tmp = x + (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.95e+140], N[Not[LessEqual[x, 6.8e-12]], $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 < -1.94999999999999987e140 or 6.8000000000000001e-12 < 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 99.1%

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

      3. associate-*l* 99.1%

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

      4. fma-def 99.1%

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

      5. pow2 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in x around inf 92.3%

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

      1. pow-base-1 92.3%

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

      2. *-lft-identity 92.3%

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

   6. Simplified 92.3%

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

   if -1.94999999999999987e140 < x < 6.8000000000000001e-12

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 90.7%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+140} \lor \neg \left(x \leq 6.8 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \]

Alternative 5: 75.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-5} \lor \neg \left(y \leq 0.0064\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot y + \left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7e-5) (not (<= y 0.0064)))
   (* z (sin y))
   (+ (* z y) (+ x (* -0.5 (* x (* y y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7e-5) || !(y <= 0.0064)) {
		tmp = z * sin(y);
	} else {
		tmp = (z * y) + (x + (-0.5 * (x * (y * 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 <= (-7d-5)) .or. (.not. (y <= 0.0064d0))) then
        tmp = z * sin(y)
    else
        tmp = (z * y) + (x + ((-0.5d0) * (x * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7e-5) || !(y <= 0.0064)) {
		tmp = z * Math.sin(y);
	} else {
		tmp = (z * y) + (x + (-0.5 * (x * (y * y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7e-5) or not (y <= 0.0064):
		tmp = z * math.sin(y)
	else:
		tmp = (z * y) + (x + (-0.5 * (x * (y * y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7e-5) || !(y <= 0.0064))
		tmp = Float64(z * sin(y));
	else
		tmp = Float64(Float64(z * y) + Float64(x + Float64(-0.5 * Float64(x * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -7e-5) || ~((y <= 0.0064)))
		tmp = z * sin(y);
	else
		tmp = (z * y) + (x + (-0.5 * (x * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7e-5], N[Not[LessEqual[y, 0.0064]], $MachinePrecision]], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] + N[(x + N[(-0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.9999999999999994e-5 or 0.00640000000000000031 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 47.9%

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

   if -6.9999999999999994e-5 < y < 0.00640000000000000031

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. expm1-log1p-u 92.9%

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

      2. expm1-udef 92.8%

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

      3. *-commutative 92.8%

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

      4. unpow2 92.8%

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

   4. Applied egg-rr 92.8%

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

      1. expm1-def 92.9%

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

      2. expm1-log1p 100.0%

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

      3. unpow2 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow2 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-5} \lor \neg \left(y \leq 0.0064\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot y + \left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]

Alternative 6: 42.3% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+165}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+150}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e+165) (* z y) (if (<= z 3.2e+150) x (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+165) {
		tmp = z * y;
	} else if (z <= 3.2e+150) {
		tmp = x;
	} else {
		tmp = z * 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 (z <= (-1d+165)) then
        tmp = z * y
    else if (z <= 3.2d+150) then
        tmp = x
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+165) {
		tmp = z * y;
	} else if (z <= 3.2e+150) {
		tmp = x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1e+165:
		tmp = z * y
	elif z <= 3.2e+150:
		tmp = x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1e+165)
		tmp = Float64(z * y);
	elseif (z <= 3.2e+150)
		tmp = x;
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1e+165)
		tmp = z * y;
	elseif (z <= 3.2e+150)
		tmp = x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1e+165], N[(z * y), $MachinePrecision], If[LessEqual[z, 3.2e+150], x, N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999899e164 or 3.20000000000000016e150 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 62.0%

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

   3. Taylor expanded in z around inf 46.2%

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

   if -9.99999999999999899e164 < z < 3.20000000000000016e150

   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 99.4%

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

      3. associate-*l* 99.4%

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

      4. fma-def 99.4%

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

      5. pow2 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in y around 0 48.4%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+165}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+150}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]

Alternative 7: 52.4% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, y, z):
	return x + (z * y)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0 58.1%

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

3. Final simplification 58.1%

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

Alternative 8: 39.6% 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 99.9%

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

   1. *-commutative 99.9%

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

   2. add-cube-cbrt 99.5%

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

   3. associate-*l* 99.5%

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

   4. fma-def 99.5%

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

   5. pow2 99.5%

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

3. Applied egg-rr 99.5%

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

4. Taylor expanded in y around 0 41.8%

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

5. Final simplification 41.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023268 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))