Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B

Percentage Accurate: 99.8% → 99.8%

Time: 9.5s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-def 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \sin y + z \cdot \cos y \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto z \cdot \cos y + x \cdot \sin y \]
4. Add Preprocessing

Alternative 3: 75.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := x \cdot \sin y\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+280}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.0095:\\ \;\;\;\;z + x \cdot y\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+67} \lor \neg \left(y \leq 1.7 \cdot 10^{+193}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))) (t_1 (* x (sin y))))
   (if (<= y -6.2e+280)
     t_0
     (if (<= y -3.3e+185)
       t_1
       (if (<= y -1.05e+18)
         t_0
         (if (<= y -2.25e-5)
           t_1
           (if (<= y 0.0095)
             (+ z (* x y))
             (if (or (<= y 5.6e+67) (not (<= y 1.7e+193))) t_1 t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double t_1 = x * sin(y);
	double tmp;
	if (y <= -6.2e+280) {
		tmp = t_0;
	} else if (y <= -3.3e+185) {
		tmp = t_1;
	} else if (y <= -1.05e+18) {
		tmp = t_0;
	} else if (y <= -2.25e-5) {
		tmp = t_1;
	} else if (y <= 0.0095) {
		tmp = z + (x * y);
	} else if ((y <= 5.6e+67) || !(y <= 1.7e+193)) {
		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 * cos(y)
    t_1 = x * sin(y)
    if (y <= (-6.2d+280)) then
        tmp = t_0
    else if (y <= (-3.3d+185)) then
        tmp = t_1
    else if (y <= (-1.05d+18)) then
        tmp = t_0
    else if (y <= (-2.25d-5)) then
        tmp = t_1
    else if (y <= 0.0095d0) then
        tmp = z + (x * y)
    else if ((y <= 5.6d+67) .or. (.not. (y <= 1.7d+193))) 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.cos(y);
	double t_1 = x * Math.sin(y);
	double tmp;
	if (y <= -6.2e+280) {
		tmp = t_0;
	} else if (y <= -3.3e+185) {
		tmp = t_1;
	} else if (y <= -1.05e+18) {
		tmp = t_0;
	} else if (y <= -2.25e-5) {
		tmp = t_1;
	} else if (y <= 0.0095) {
		tmp = z + (x * y);
	} else if ((y <= 5.6e+67) || !(y <= 1.7e+193)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	t_1 = x * math.sin(y)
	tmp = 0
	if y <= -6.2e+280:
		tmp = t_0
	elif y <= -3.3e+185:
		tmp = t_1
	elif y <= -1.05e+18:
		tmp = t_0
	elif y <= -2.25e-5:
		tmp = t_1
	elif y <= 0.0095:
		tmp = z + (x * y)
	elif (y <= 5.6e+67) or not (y <= 1.7e+193):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	t_1 = Float64(x * sin(y))
	tmp = 0.0
	if (y <= -6.2e+280)
		tmp = t_0;
	elseif (y <= -3.3e+185)
		tmp = t_1;
	elseif (y <= -1.05e+18)
		tmp = t_0;
	elseif (y <= -2.25e-5)
		tmp = t_1;
	elseif (y <= 0.0095)
		tmp = Float64(z + Float64(x * y));
	elseif ((y <= 5.6e+67) || !(y <= 1.7e+193))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	t_1 = x * sin(y);
	tmp = 0.0;
	if (y <= -6.2e+280)
		tmp = t_0;
	elseif (y <= -3.3e+185)
		tmp = t_1;
	elseif (y <= -1.05e+18)
		tmp = t_0;
	elseif (y <= -2.25e-5)
		tmp = t_1;
	elseif (y <= 0.0095)
		tmp = z + (x * y);
	elseif ((y <= 5.6e+67) || ~((y <= 1.7e+193)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.2e+280], t$95$0, If[LessEqual[y, -3.3e+185], t$95$1, If[LessEqual[y, -1.05e+18], t$95$0, If[LessEqual[y, -2.25e-5], t$95$1, If[LessEqual[y, 0.0095], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 5.6e+67], N[Not[LessEqual[y, 1.7e+193]], $MachinePrecision]], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.1999999999999999e280 or -3.30000000000000011e185 < y < -1.05e18 or 5.5999999999999995e67 < y < 1.69999999999999993e193

   1. Initial program 99.5%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 69.8%

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

   if -6.1999999999999999e280 < y < -3.30000000000000011e185 or -1.05e18 < y < -2.25000000000000014e-5 or 0.00949999999999999976 < y < 5.5999999999999995e67 or 1.69999999999999993e193 < y

   1. Initial program 99.7%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 71.5%

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

   if -2.25000000000000014e-5 < y < 0.00949999999999999976

   1. Initial program 100.0%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.8%

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

      1. *-commutative 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+280}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{+185}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{+18}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq 0.0095:\\ \;\;\;\;z + x \cdot y\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+67} \lor \neg \left(y \leq 1.7 \cdot 10^{+193}\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := x \cdot \sin y\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{+275}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5 \cdot 10^{+181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.00115:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+69} \lor \neg \left(y \leq 1.6 \cdot 10^{+196}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))) (t_1 (* x (sin y))))
   (if (<= y -5.4e+275)
     t_0
     (if (<= y -5e+181)
       t_1
       (if (<= y -5.7e+17)
         t_0
         (if (<= y -2.7e-5)
           t_1
           (if (<= y 0.00115)
             (fma y x z)
             (if (or (<= y 1.6e+69) (not (<= y 1.6e+196))) t_1 t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double t_1 = x * sin(y);
	double tmp;
	if (y <= -5.4e+275) {
		tmp = t_0;
	} else if (y <= -5e+181) {
		tmp = t_1;
	} else if (y <= -5.7e+17) {
		tmp = t_0;
	} else if (y <= -2.7e-5) {
		tmp = t_1;
	} else if (y <= 0.00115) {
		tmp = fma(y, x, z);
	} else if ((y <= 1.6e+69) || !(y <= 1.6e+196)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	t_1 = Float64(x * sin(y))
	tmp = 0.0
	if (y <= -5.4e+275)
		tmp = t_0;
	elseif (y <= -5e+181)
		tmp = t_1;
	elseif (y <= -5.7e+17)
		tmp = t_0;
	elseif (y <= -2.7e-5)
		tmp = t_1;
	elseif (y <= 0.00115)
		tmp = fma(y, x, z);
	elseif ((y <= 1.6e+69) || !(y <= 1.6e+196))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.4e+275], t$95$0, If[LessEqual[y, -5e+181], t$95$1, If[LessEqual[y, -5.7e+17], t$95$0, If[LessEqual[y, -2.7e-5], t$95$1, If[LessEqual[y, 0.00115], N[(y * x + z), $MachinePrecision], If[Or[LessEqual[y, 1.6e+69], N[Not[LessEqual[y, 1.6e+196]], $MachinePrecision]], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.40000000000000031e275 or -5.0000000000000003e181 < y < -5.7e17 or 1.59999999999999992e69 < y < 1.59999999999999996e196

   1. Initial program 99.5%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 69.8%

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

   if -5.40000000000000031e275 < y < -5.0000000000000003e181 or -5.7e17 < y < -2.6999999999999999e-5 or 0.00115 < y < 1.59999999999999992e69 or 1.59999999999999996e196 < y

   1. Initial program 99.7%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 71.5%

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

   if -2.6999999999999999e-5 < y < 0.00115

   1. Initial program 100.0%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. fma-def 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+275}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq -5 \cdot 10^{+181}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{+17}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq 0.00115:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+69} \lor \neg \left(y \leq 1.6 \cdot 10^{+196}\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-49} \lor \neg \left(x \leq 1.55 \cdot 10^{-87}\right):\\ \;\;\;\;z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5e-49) (not (<= x 1.55e-87)))
   (+ z (* x (sin y)))
   (* z (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5e-49) || !(x <= 1.55e-87)) {
		tmp = z + (x * sin(y));
	} else {
		tmp = z * 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 ((x <= (-5d-49)) .or. (.not. (x <= 1.55d-87))) then
        tmp = z + (x * sin(y))
    else
        tmp = z * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5e-49) || !(x <= 1.55e-87)) {
		tmp = z + (x * Math.sin(y));
	} else {
		tmp = z * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5e-49) or not (x <= 1.55e-87):
		tmp = z + (x * math.sin(y))
	else:
		tmp = z * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5e-49) || !(x <= 1.55e-87))
		tmp = Float64(z + Float64(x * sin(y)));
	else
		tmp = Float64(z * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5e-49) || ~((x <= 1.55e-87)))
		tmp = z + (x * sin(y));
	else
		tmp = z * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5e-49], N[Not[LessEqual[x, 1.55e-87]], $MachinePrecision]], N[(z + N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999999e-49 or 1.54999999999999999e-87 < x

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 92.0%

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

   if -4.9999999999999999e-49 < x < 1.54999999999999999e-87

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 85.3%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-49} \lor \neg \left(x \leq 1.55 \cdot 10^{-87}\right):\\ \;\;\;\;z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-5} \lor \neg \left(y \leq 0.0225\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.25e-5) (not (<= y 0.0225))) (* x (sin y)) (+ z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.25e-5) || !(y <= 0.0225)) {
		tmp = x * sin(y);
	} else {
		tmp = z + (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 <= (-1.25d-5)) .or. (.not. (y <= 0.0225d0))) then
        tmp = x * sin(y)
    else
        tmp = z + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.25e-5) || !(y <= 0.0225)) {
		tmp = x * Math.sin(y);
	} else {
		tmp = z + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.25e-5) or not (y <= 0.0225):
		tmp = x * math.sin(y)
	else:
		tmp = z + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.25e-5) || !(y <= 0.0225))
		tmp = Float64(x * sin(y));
	else
		tmp = Float64(z + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.25e-5) || ~((y <= 0.0225)))
		tmp = x * sin(y);
	else
		tmp = z + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.25e-5], N[Not[LessEqual[y, 0.0225]], $MachinePrecision]], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25000000000000006e-5 or 0.022499999999999999 < y

   1. Initial program 99.6%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 50.6%

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

   if -1.25000000000000006e-5 < y < 0.022499999999999999

   1. Initial program 100.0%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.8%

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

      1. *-commutative 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-5} \lor \neg \left(y \leq 0.0225\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.2% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-133}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-103}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.5e-133) z (if (<= z 2.8e-103) (* x y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.5e-133) {
		tmp = z;
	} else if (z <= 2.8e-103) {
		tmp = x * y;
	} else {
		tmp = 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 (z <= (-7.5d-133)) then
        tmp = z
    else if (z <= 2.8d-103) then
        tmp = x * y
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.5e-133) {
		tmp = z;
	} else if (z <= 2.8e-103) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.5e-133:
		tmp = z
	elif z <= 2.8e-103:
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.5e-133)
		tmp = z;
	elseif (z <= 2.8e-103)
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7.5e-133)
		tmp = z;
	elseif (z <= 2.8e-103)
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.5e-133], z, If[LessEqual[z, 2.8e-103], N[(x * y), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -7.4999999999999999e-133 or 2.80000000000000023e-103 < z

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. fma-def 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0 47.2%

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

   if -7.4999999999999999e-133 < z < 2.80000000000000023e-103

   1. Initial program 99.9%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 59.5%

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

      1. *-commutative 59.5%

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

   5. Simplified 59.5%

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

   6. Taylor expanded in z around 0 42.3%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-133}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-103}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.5% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \sin y + z \cdot \cos y \]
2. Add Preprocessing

3. Taylor expanded in y around 0 55.9%

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

   1. *-commutative 55.9%

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

5. Simplified 55.9%

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

6. Final simplification 55.9%

   \[\leadsto z + x \cdot y \]
7. Add Preprocessing

Alternative 9: 39.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x, y, z)
	return z
end

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. fma-def 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in y around 0 38.3%

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

6. Final simplification 38.3%

   \[\leadsto z \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))