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

Percentage Accurate: 99.8% → 99.8%

Time: 7.9s

Alternatives: 8

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. Final simplification 99.8%

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

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. Final simplification 99.8%

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

Alternative 3: 74.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{-59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-141}:\\ \;\;\;\;z + x \cdot y\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-143} \lor \neg \left(z \leq 5.8 \cdot 10^{-123}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -2.5e-59)
     t_0
     (if (<= z -3.2e-141)
       (+ z (* x y))
       (if (or (<= z -5.5e-143) (not (<= z 5.8e-123))) t_0 (* x (sin y)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -2.5e-59) {
		tmp = t_0;
	} else if (z <= -3.2e-141) {
		tmp = z + (x * y);
	} else if ((z <= -5.5e-143) || !(z <= 5.8e-123)) {
		tmp = t_0;
	} else {
		tmp = x * 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) :: t_0
    real(8) :: tmp
    t_0 = z * cos(y)
    if (z <= (-2.5d-59)) then
        tmp = t_0
    else if (z <= (-3.2d-141)) then
        tmp = z + (x * y)
    else if ((z <= (-5.5d-143)) .or. (.not. (z <= 5.8d-123))) then
        tmp = t_0
    else
        tmp = x * sin(y)
    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 tmp;
	if (z <= -2.5e-59) {
		tmp = t_0;
	} else if (z <= -3.2e-141) {
		tmp = z + (x * y);
	} else if ((z <= -5.5e-143) || !(z <= 5.8e-123)) {
		tmp = t_0;
	} else {
		tmp = x * Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -2.5e-59:
		tmp = t_0
	elif z <= -3.2e-141:
		tmp = z + (x * y)
	elif (z <= -5.5e-143) or not (z <= 5.8e-123):
		tmp = t_0
	else:
		tmp = x * math.sin(y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -2.5e-59)
		tmp = t_0;
	elseif (z <= -3.2e-141)
		tmp = Float64(z + Float64(x * y));
	elseif ((z <= -5.5e-143) || !(z <= 5.8e-123))
		tmp = t_0;
	else
		tmp = Float64(x * sin(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	tmp = 0.0;
	if (z <= -2.5e-59)
		tmp = t_0;
	elseif (z <= -3.2e-141)
		tmp = z + (x * y);
	elseif ((z <= -5.5e-143) || ~((z <= 5.8e-123)))
		tmp = t_0;
	else
		tmp = x * sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.5e-59], t$95$0, If[LessEqual[z, -3.2e-141], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -5.5e-143], N[Not[LessEqual[z, 5.8e-123]], $MachinePrecision]], t$95$0, N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.5000000000000001e-59 or -3.2000000000000001e-141 < z < -5.50000000000000041e-143 or 5.80000000000000007e-123 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 80.6%

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

   if -2.5000000000000001e-59 < z < -3.2000000000000001e-141

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 80.0%

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

      1. *-commutative 80.0%

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

   4. Simplified 80.0%

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

   if -5.50000000000000041e-143 < z < 5.80000000000000007e-123

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 80.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-59}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-141}:\\ \;\;\;\;z + x \cdot y\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-143} \lor \neg \left(z \leq 5.8 \cdot 10^{-123}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \]

Alternative 4: 74.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-143} \lor \neg \left(z \leq 9 \cdot 10^{-125}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -3.5e-58)
     t_0
     (if (<= z -3.2e-141)
       (fma y x z)
       (if (or (<= z -7e-143) (not (<= z 9e-125))) t_0 (* x (sin y)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -3.5e-58) {
		tmp = t_0;
	} else if (z <= -3.2e-141) {
		tmp = fma(y, x, z);
	} else if ((z <= -7e-143) || !(z <= 9e-125)) {
		tmp = t_0;
	} else {
		tmp = x * sin(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -3.5e-58)
		tmp = t_0;
	elseif (z <= -3.2e-141)
		tmp = fma(y, x, z);
	elseif ((z <= -7e-143) || !(z <= 9e-125))
		tmp = t_0;
	else
		tmp = Float64(x * sin(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.5e-58], t$95$0, If[LessEqual[z, -3.2e-141], N[(y * x + z), $MachinePrecision], If[Or[LessEqual[z, -7e-143], N[Not[LessEqual[z, 9e-125]], $MachinePrecision]], t$95$0, N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.4999999999999999e-58 or -3.2000000000000001e-141 < z < -7.00000000000000011e-143 or 9.00000000000000024e-125 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 80.6%

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

   if -3.4999999999999999e-58 < z < -3.2000000000000001e-141

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 80.0%

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

      1. +-commutative 80.0%

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

      2. *-commutative 80.0%

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

      3. fma-def 80.0%

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

   4. Simplified 80.0%

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

   if -7.00000000000000011e-143 < z < 9.00000000000000024e-125

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 80.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-58}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-143} \lor \neg \left(z \leq 9 \cdot 10^{-125}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \]

Alternative 5: 85.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-83} \lor \neg \left(x \leq 6.4 \cdot 10^{-149}\right):\\ \;\;\;\;z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.1e-83) (not (<= x 6.4e-149)))
   (+ z (* x (sin y)))
   (* z (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.1e-83) || !(x <= 6.4e-149)) {
		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 <= (-1.1d-83)) .or. (.not. (x <= 6.4d-149))) 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 <= -1.1e-83) || !(x <= 6.4e-149)) {
		tmp = z + (x * Math.sin(y));
	} else {
		tmp = z * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.1e-83) or not (x <= 6.4e-149):
		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 <= -1.1e-83) || !(x <= 6.4e-149))
		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 <= -1.1e-83) || ~((x <= 6.4e-149)))
		tmp = z + (x * sin(y));
	else
		tmp = z * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.1e-83], N[Not[LessEqual[x, 6.4e-149]], $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 < -1.10000000000000004e-83 or 6.40000000000000004e-149 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 88.7%

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

   if -1.10000000000000004e-83 < x < 6.40000000000000004e-149

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 92.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-83} \lor \neg \left(x \leq 6.4 \cdot 10^{-149}\right):\\ \;\;\;\;z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]

Alternative 6: 75.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00082 \lor \neg \left(y \leq 2.85 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.00082) (not (<= y 2.85e-5)))
   (* x (sin y))
   (+ z (* y (+ x (* y (* z -0.5)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00082) || !(y <= 2.85e-5)) {
		tmp = x * sin(y);
	} else {
		tmp = z + (y * (x + (y * (z * -0.5))));
	}
	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 <= (-0.00082d0)) .or. (.not. (y <= 2.85d-5))) then
        tmp = x * sin(y)
    else
        tmp = z + (y * (x + (y * (z * (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00082) || !(y <= 2.85e-5)) {
		tmp = x * Math.sin(y);
	} else {
		tmp = z + (y * (x + (y * (z * -0.5))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.00082) or not (y <= 2.85e-5):
		tmp = x * math.sin(y)
	else:
		tmp = z + (y * (x + (y * (z * -0.5))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.00082) || !(y <= 2.85e-5))
		tmp = Float64(x * sin(y));
	else
		tmp = Float64(z + Float64(y * Float64(x + Float64(y * Float64(z * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.00082) || ~((y <= 2.85e-5)))
		tmp = x * sin(y);
	else
		tmp = z + (y * (x + (y * (z * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.00082], N[Not[LessEqual[y, 2.85e-5]], $MachinePrecision]], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(z + N[(y * N[(x + N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.1999999999999998e-4 or 2.8500000000000002e-5 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf 49.2%

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

   if -8.1999999999999998e-4 < y < 2.8500000000000002e-5

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. add-sqr-sqrt 53.2%

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

      4. associate-*r* 53.2%

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

      5. fma-def 53.2%

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

   3. Applied egg-rr 53.2%

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

   4. Taylor expanded in y around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. metadata-eval 99.9%

         \[\leadsto z + \left({y}^{2} \cdot \left(z \cdot \color{blue}{\left(-0.3333333333333333 + -0.16666666666666666\right)}\right) + x \cdot y\right) \]

      4. distribute-rgt-out 99.9%

         \[\leadsto z + \left({y}^{2} \cdot \color{blue}{\left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right)} + x \cdot y\right) \]

      5. unpow2 99.9%

         \[\leadsto z + \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right) + x \cdot y\right) \]

      6. associate-*l* 99.9%

         \[\leadsto z + \left(\color{blue}{y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right)\right)} + x \cdot y\right) \]

      7. *-commutative 99.9%

         \[\leadsto z + \left(y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right)\right) + \color{blue}{y \cdot x}\right) \]

      8. distribute-lft-out 99.9%

         \[\leadsto z + \color{blue}{y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right) + x\right)} \]

      9. distribute-rgt-out 99.9%

         \[\leadsto z + y \cdot \left(y \cdot \color{blue}{\left(z \cdot \left(-0.3333333333333333 + -0.16666666666666666\right)\right)} + x\right) \]

      10. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00082 \lor \neg \left(y \leq 2.85 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 7: 52.6% 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. Taylor expanded in y around 0 58.0%

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

   1. *-commutative 58.0%

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

4. Simplified 58.0%

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

5. Final simplification 58.0%

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

Alternative 8: 16.8% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in y around 0 58.0%

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

   1. *-commutative 58.0%

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

4. Simplified 58.0%

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

5. Taylor expanded in z around 0 20.6%

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

   1. *-commutative 20.6%

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

7. Simplified 20.6%

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

8. Final simplification 20.6%

   \[\leadsto x \cdot y \]

Reproduce

herbie shell --seed 2023271 
(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))))