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

Percentage Accurate: 99.8% → 99.8%

Time: 8.5s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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. fma-def 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 4: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot z\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -8.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2700:\\ \;\;\;\;y \cdot z + \left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+118} \lor \neg \left(y \leq 2.4 \cdot 10^{+176}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) z)) (t_1 (* x (cos y))))
   (if (<= y -5.5e+207)
     t_0
     (if (<= y -7e+156)
       t_1
       (if (<= y -4.8e+113)
         t_0
         (if (<= y -8.5)
           t_1
           (if (<= y 2700.0)
             (+ (* y z) (+ x (* -0.5 (* x (* y y)))))
             (if (or (<= y 2e+118) (not (<= y 2.4e+176))) t_0 t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * z;
	double t_1 = x * cos(y);
	double tmp;
	if (y <= -5.5e+207) {
		tmp = t_0;
	} else if (y <= -7e+156) {
		tmp = t_1;
	} else if (y <= -4.8e+113) {
		tmp = t_0;
	} else if (y <= -8.5) {
		tmp = t_1;
	} else if (y <= 2700.0) {
		tmp = (y * z) + (x + (-0.5 * (x * (y * y))));
	} else if ((y <= 2e+118) || !(y <= 2.4e+176)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = sin(y) * z
    t_1 = x * cos(y)
    if (y <= (-5.5d+207)) then
        tmp = t_0
    else if (y <= (-7d+156)) then
        tmp = t_1
    else if (y <= (-4.8d+113)) then
        tmp = t_0
    else if (y <= (-8.5d0)) then
        tmp = t_1
    else if (y <= 2700.0d0) then
        tmp = (y * z) + (x + ((-0.5d0) * (x * (y * y))))
    else if ((y <= 2d+118) .or. (.not. (y <= 2.4d+176))) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) * z;
	double t_1 = x * Math.cos(y);
	double tmp;
	if (y <= -5.5e+207) {
		tmp = t_0;
	} else if (y <= -7e+156) {
		tmp = t_1;
	} else if (y <= -4.8e+113) {
		tmp = t_0;
	} else if (y <= -8.5) {
		tmp = t_1;
	} else if (y <= 2700.0) {
		tmp = (y * z) + (x + (-0.5 * (x * (y * y))));
	} else if ((y <= 2e+118) || !(y <= 2.4e+176)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * z
	t_1 = x * math.cos(y)
	tmp = 0
	if y <= -5.5e+207:
		tmp = t_0
	elif y <= -7e+156:
		tmp = t_1
	elif y <= -4.8e+113:
		tmp = t_0
	elif y <= -8.5:
		tmp = t_1
	elif y <= 2700.0:
		tmp = (y * z) + (x + (-0.5 * (x * (y * y))))
	elif (y <= 2e+118) or not (y <= 2.4e+176):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * z)
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -5.5e+207)
		tmp = t_0;
	elseif (y <= -7e+156)
		tmp = t_1;
	elseif (y <= -4.8e+113)
		tmp = t_0;
	elseif (y <= -8.5)
		tmp = t_1;
	elseif (y <= 2700.0)
		tmp = Float64(Float64(y * z) + Float64(x + Float64(-0.5 * Float64(x * Float64(y * y)))));
	elseif ((y <= 2e+118) || !(y <= 2.4e+176))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * z;
	t_1 = x * cos(y);
	tmp = 0.0;
	if (y <= -5.5e+207)
		tmp = t_0;
	elseif (y <= -7e+156)
		tmp = t_1;
	elseif (y <= -4.8e+113)
		tmp = t_0;
	elseif (y <= -8.5)
		tmp = t_1;
	elseif (y <= 2700.0)
		tmp = (y * z) + (x + (-0.5 * (x * (y * y))));
	elseif ((y <= 2e+118) || ~((y <= 2.4e+176)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e+207], t$95$0, If[LessEqual[y, -7e+156], t$95$1, If[LessEqual[y, -4.8e+113], t$95$0, If[LessEqual[y, -8.5], t$95$1, If[LessEqual[y, 2700.0], N[(N[(y * z), $MachinePrecision] + N[(x + N[(-0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 2e+118], N[Not[LessEqual[y, 2.4e+176]], $MachinePrecision]], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.50000000000000036e207 or -7.0000000000000006e156 < y < -4.79999999999999966e113 or 2700 < y < 1.99999999999999993e118 or 2.4000000000000001e176 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 64.4%

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

   if -5.50000000000000036e207 < y < -7.0000000000000006e156 or -4.79999999999999966e113 < y < -8.5 or 1.99999999999999993e118 < y < 2.4000000000000001e176

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in z around 0 70.3%

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

   if -8.5 < y < 2700

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.6%

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

      1. expm1-log1p-u 95.6%

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

      2. expm1-udef 95.6%

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

      3. unpow2 95.6%

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

      4. associate-*l* 95.6%

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

   4. Applied egg-rr 95.6%

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

      1. expm1-def 95.6%

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

      2. expm1-log1p 98.6%

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

      3. associate-*r* 98.6%

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

      4. *-commutative 98.6%

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

   6. Simplified 98.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+207}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{elif}\;y \leq -7 \cdot 10^{+156}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+113}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{elif}\;y \leq -8.5:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 2700:\\ \;\;\;\;y \cdot z + \left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+118} \lor \neg \left(y \leq 2.4 \cdot 10^{+176}\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 5: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot z\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+208}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -8.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+114} \lor \neg \left(y \leq 1.3 \cdot 10^{+172}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) z)) (t_1 (* x (cos y))))
   (if (<= y -1.25e+208)
     t_0
     (if (<= y -7e+156)
       t_1
       (if (<= y -1.4e+106)
         t_0
         (if (<= y -8.5)
           t_1
           (if (<= y 3.8e-11)
             (fma y z x)
             (if (or (<= y 3.4e+114) (not (<= y 1.3e+172))) t_0 t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * z;
	double t_1 = x * cos(y);
	double tmp;
	if (y <= -1.25e+208) {
		tmp = t_0;
	} else if (y <= -7e+156) {
		tmp = t_1;
	} else if (y <= -1.4e+106) {
		tmp = t_0;
	} else if (y <= -8.5) {
		tmp = t_1;
	} else if (y <= 3.8e-11) {
		tmp = fma(y, z, x);
	} else if ((y <= 3.4e+114) || !(y <= 1.3e+172)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * z)
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -1.25e+208)
		tmp = t_0;
	elseif (y <= -7e+156)
		tmp = t_1;
	elseif (y <= -1.4e+106)
		tmp = t_0;
	elseif (y <= -8.5)
		tmp = t_1;
	elseif (y <= 3.8e-11)
		tmp = fma(y, z, x);
	elseif ((y <= 3.4e+114) || !(y <= 1.3e+172))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.25e+208], t$95$0, If[LessEqual[y, -7e+156], t$95$1, If[LessEqual[y, -1.4e+106], t$95$0, If[LessEqual[y, -8.5], t$95$1, If[LessEqual[y, 3.8e-11], N[(y * z + x), $MachinePrecision], If[Or[LessEqual[y, 3.4e+114], N[Not[LessEqual[y, 1.3e+172]], $MachinePrecision]], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2500000000000001e208 or -7.0000000000000006e156 < y < -1.39999999999999996e106 or 3.7999999999999998e-11 < y < 3.4000000000000001e114 or 1.3e172 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 64.6%

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

   if -1.2500000000000001e208 < y < -7.0000000000000006e156 or -1.39999999999999996e106 < y < -8.5 or 3.4000000000000001e114 < y < 1.3e172

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in z around 0 70.3%

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

   if -8.5 < y < 3.7999999999999998e-11

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.3%

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

      1. fma-def 99.3%

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

   4. Simplified 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, z, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+208}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{elif}\;y \leq -7 \cdot 10^{+156}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{+106}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{elif}\;y \leq -8.5:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+114} \lor \neg \left(y \leq 1.3 \cdot 10^{+172}\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 6: 86.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-69} \lor \neg \left(z \leq 6.6 \cdot 10^{-40}\right):\\ \;\;\;\;x + \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.1e-69) (not (<= z 6.6e-40)))
   (+ x (* (sin y) z))
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1e-69) || !(z <= 6.6e-40)) {
		tmp = x + (sin(y) * z);
	} 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) :: tmp
    if ((z <= (-1.1d-69)) .or. (.not. (z <= 6.6d-40))) then
        tmp = x + (sin(y) * z)
    else
        tmp = x * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1e-69) || !(z <= 6.6e-40)) {
		tmp = x + (Math.sin(y) * z);
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.1e-69) or not (z <= 6.6e-40):
		tmp = x + (math.sin(y) * z)
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.1e-69) || !(z <= 6.6e-40))
		tmp = Float64(x + Float64(sin(y) * z));
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.1e-69) || ~((z <= 6.6e-40)))
		tmp = x + (sin(y) * z);
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.1e-69], N[Not[LessEqual[z, 6.6e-40]], $MachinePrecision]], N[(x + N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1e-69 or 6.59999999999999986e-40 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 90.8%

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

   if -1.1e-69 < z < 6.59999999999999986e-40

   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. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in z around 0 88.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-69} \lor \neg \left(z \leq 6.6 \cdot 10^{-40}\right):\\ \;\;\;\;x + \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 7: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.065 \lor \neg \left(y \leq 2700\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot z + \left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.065) (not (<= y 2700.0)))
   (* (sin y) z)
   (+ (* y z) (+ x (* -0.5 (* x (* y y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.065) || !(y <= 2700.0)) {
		tmp = sin(y) * z;
	} else {
		tmp = (y * z) + (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 <= (-0.065d0)) .or. (.not. (y <= 2700.0d0))) then
        tmp = sin(y) * z
    else
        tmp = (y * z) + (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 <= -0.065) || !(y <= 2700.0)) {
		tmp = Math.sin(y) * z;
	} else {
		tmp = (y * z) + (x + (-0.5 * (x * (y * y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.065) or not (y <= 2700.0):
		tmp = math.sin(y) * z
	else:
		tmp = (y * z) + (x + (-0.5 * (x * (y * y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.065) || !(y <= 2700.0))
		tmp = Float64(sin(y) * z);
	else
		tmp = Float64(Float64(y * z) + 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 <= -0.065) || ~((y <= 2700.0)))
		tmp = sin(y) * z;
	else
		tmp = (y * z) + (x + (-0.5 * (x * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.065], N[Not[LessEqual[y, 2700.0]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision], N[(N[(y * z), $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 < -0.065000000000000002 or 2700 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 50.8%

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

   if -0.065000000000000002 < y < 2700

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.3%

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

      1. expm1-log1p-u 96.2%

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

      2. expm1-udef 96.2%

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

      3. unpow2 96.2%

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

      4. associate-*l* 96.2%

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

   4. Applied egg-rr 96.2%

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

      1. expm1-def 96.2%

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

      2. expm1-log1p 99.3%

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

      3. associate-*r* 99.3%

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

      4. *-commutative 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.065 \lor \neg \left(y \leq 2700\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot z + \left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]

Alternative 8: 41.2% accurate, 40.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+220}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z -1.15e+220) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15e+220) {
		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 <= (-1.15d+220)) 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 <= -1.15e+220) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.15e+220:
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.15e+220)
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.15e+220)
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.15e+220], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.14999999999999998e220

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 60.6%

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

   3. Taylor expanded in y around inf 45.8%

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

   if -1.14999999999999998e220 < 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. *-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 45.1%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+220}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 53.4% 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 53.9%

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

3. Final simplification 53.9%

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

Alternative 10: 39.9% 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. fma-def 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in y around 0 42.8%

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

5. Final simplification 42.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023214 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))