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

Percentage Accurate: 99.8% → 99.8%

Time: 8.0s

Alternatives: 11

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: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+149} \lor \neg \left(x \leq 2.2 \cdot 10^{+112}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.95e+149) (not (<= x 2.2e+112)))
   (* x (cos y))
   (fma (sin y) z x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.95e+149) || !(x <= 2.2e+112)) {
		tmp = x * cos(y);
	} else {
		tmp = fma(sin(y), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.95e+149) || !(x <= 2.2e+112))
		tmp = Float64(x * cos(y));
	else
		tmp = fma(sin(y), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.95e+149], N[Not[LessEqual[x, 2.2e+112]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.95e149 or 2.1999999999999999e112 < x

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      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 92.8%

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

   if -1.95e149 < x < 2.1999999999999999e112

   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 87.4%

      \[\leadsto \mathsf{fma}\left(\sin y, z, \color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+149} \lor \neg \left(x \leq 2.2 \cdot 10^{+112}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, x\right)\\ \end{array} \]

Alternative 4: 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 5: 85.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+149} \lor \neg \left(x \leq 8.5 \cdot 10^{+113}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + \sin y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.95e+149) (not (<= x 8.5e+113)))
   (* x (cos y))
   (+ x (* (sin y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.95e+149) || !(x <= 8.5e+113)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (sin(y) * 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 ((x <= (-1.95d+149)) .or. (.not. (x <= 8.5d+113))) then
        tmp = x * cos(y)
    else
        tmp = x + (sin(y) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.95e+149) || !(x <= 8.5e+113)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (Math.sin(y) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.95e+149) or not (x <= 8.5e+113):
		tmp = x * math.cos(y)
	else:
		tmp = x + (math.sin(y) * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.95e+149) || !(x <= 8.5e+113))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(sin(y) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.95e+149) || ~((x <= 8.5e+113)))
		tmp = x * cos(y);
	else
		tmp = x + (sin(y) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.95e+149], N[Not[LessEqual[x, 8.5e+113]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.95e149 or 8.5000000000000001e113 < x

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      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 92.8%

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

   if -1.95e149 < x < 8.5000000000000001e113

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 87.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+149} \lor \neg \left(x \leq 8.5 \cdot 10^{+113}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + \sin y \cdot z\\ \end{array} \]

Alternative 6: 73.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-8} \lor \neg \left(y \leq 6.5 \cdot 10^{-5}\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.6e-8) (not (<= y 6.5e-5))) (* (sin y) z) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.6e-8) || !(y <= 6.5e-5)) {
		tmp = sin(y) * z;
	} else {
		tmp = x + (y * 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 ((y <= (-5.6d-8)) .or. (.not. (y <= 6.5d-5))) then
        tmp = sin(y) * z
    else
        tmp = x + (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.6e-8) || !(y <= 6.5e-5)) {
		tmp = Math.sin(y) * z;
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.6e-8) or not (y <= 6.5e-5):
		tmp = math.sin(y) * z
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.6e-8) || !(y <= 6.5e-5))
		tmp = Float64(sin(y) * z);
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5.6e-8) || ~((y <= 6.5e-5)))
		tmp = sin(y) * z;
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.6e-8], N[Not[LessEqual[y, 6.5e-5]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5999999999999999e-8 or 6.49999999999999943e-5 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 54.8%

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

   if -5.5999999999999999e-8 < y < 6.49999999999999943e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-8} \lor \neg \left(y \leq 6.5 \cdot 10^{-5}\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 7: 74.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -65:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.000205:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -65.0)
   (* x (cos y))
   (if (<= y 0.000205) (+ x (* y z)) (* (sin y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -65.0) {
		tmp = x * cos(y);
	} else if (y <= 0.000205) {
		tmp = x + (y * z);
	} else {
		tmp = sin(y) * 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 (y <= (-65.0d0)) then
        tmp = x * cos(y)
    else if (y <= 0.000205d0) then
        tmp = x + (y * z)
    else
        tmp = sin(y) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -65.0) {
		tmp = x * Math.cos(y);
	} else if (y <= 0.000205) {
		tmp = x + (y * z);
	} else {
		tmp = Math.sin(y) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -65.0:
		tmp = x * math.cos(y)
	elif y <= 0.000205:
		tmp = x + (y * z)
	else:
		tmp = math.sin(y) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -65.0)
		tmp = Float64(x * cos(y));
	elseif (y <= 0.000205)
		tmp = Float64(x + Float64(y * z));
	else
		tmp = Float64(sin(y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -65.0)
		tmp = x * cos(y);
	elseif (y <= 0.000205)
		tmp = x + (y * z);
	else
		tmp = sin(y) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -65.0], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.000205], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -65

   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 57.7%

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

   if -65 < y < 2.05e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.4%

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

   if 2.05e-4 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 62.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -65:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.000205:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot z\\ \end{array} \]

Alternative 8: 74.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -65:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.000235:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -65.0)
   (* x (cos y))
   (if (<= y 0.000235) (fma y z x) (* (sin y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -65.0) {
		tmp = x * cos(y);
	} else if (y <= 0.000235) {
		tmp = fma(y, z, x);
	} else {
		tmp = sin(y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -65.0)
		tmp = Float64(x * cos(y));
	elseif (y <= 0.000235)
		tmp = fma(y, z, x);
	else
		tmp = Float64(sin(y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -65.0], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.000235], N[(y * z + x), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -65

   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 57.7%

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

   if -65 < y < 2.34999999999999993e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.4%

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

      1. fma-def 97.5%

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

   4. Simplified 97.5%

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

   if 2.34999999999999993e-4 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 62.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -65:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.000235:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot z\\ \end{array} \]

Alternative 9: 39.8% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+142}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.8e+142) (* y z) (if (<= z 8.6e-29) x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.8e+142) {
		tmp = y * z;
	} else if (z <= 8.6e-29) {
		tmp = x;
	} else {
		tmp = y * 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 <= (-1.8d+142)) then
        tmp = y * z
    else if (z <= 8.6d-29) then
        tmp = x
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.8e+142) {
		tmp = y * z;
	} else if (z <= 8.6e-29) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.8e+142:
		tmp = y * z
	elif z <= 8.6e-29:
		tmp = x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.8e+142)
		tmp = Float64(y * z);
	elseif (z <= 8.6e-29)
		tmp = x;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.8e+142)
		tmp = y * z;
	elseif (z <= 8.6e-29)
		tmp = x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.8e+142], N[(y * z), $MachinePrecision], If[LessEqual[z, 8.6e-29], x, N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8000000000000001e142 or 8.5999999999999996e-29 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 72.4%

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

   3. Taylor expanded in y around 0 36.9%

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

   if -1.8000000000000001e142 < z < 8.5999999999999996e-29

   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 42.7%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+142}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 10: 51.5% 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 51.0%

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

3. Final simplification 51.0%

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

Alternative 11: 38.6% 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 99.8%

   \[x \cdot \cos y + z \cdot \sin y \]
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 33.1%

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

5. Final simplification 33.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023229 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))