Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 19.6s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (sin(y) / y)
end function

Java

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

Python

def code(x, y):
	return x * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (sin(y) / y)
end function

Java

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

Python

def code(x, y):
	return x * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (sin(y) / y)
end function

Java

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

Python

def code(x, y):
	return x * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 56.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 500000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} \cdot \frac{1}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 500000.0) x (/ 1.0 (* (/ y x) (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 500000.0) {
		tmp = x;
	} else {
		tmp = 1.0 / ((y / x) * (1.0 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 500000.0d0) then
        tmp = x
    else
        tmp = 1.0d0 / ((y / x) * (1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 500000.0) {
		tmp = x;
	} else {
		tmp = 1.0 / ((y / x) * (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 500000.0:
		tmp = x
	else:
		tmp = 1.0 / ((y / x) * (1.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 500000.0)
		tmp = x;
	else
		tmp = Float64(1.0 / Float64(Float64(y / x) * Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 500000.0)
		tmp = x;
	else
		tmp = 1.0 / ((y / x) * (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 500000.0], x, N[(1.0 / N[(N[(y / x), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5e5

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 61.3%

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

   if 5e5 < y

   1. Initial program 99.5%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 99.5%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{\sin y}}} \]

      2. un-div-inv 99.5%

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

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around 0 25.9%

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

      1. clear-num 27.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \cdot y \]

      2. remove-double-div 27.5%

         \[\leadsto \frac{1}{\frac{y}{x}} \cdot \color{blue}{\frac{1}{\frac{1}{y}}} \]

      3. frac-times 27.5%

         \[\leadsto \color{blue}{\frac{1 \cdot 1}{\frac{y}{x} \cdot \frac{1}{y}}} \]

      4. metadata-eval 27.5%

         \[\leadsto \frac{\color{blue}{1}}{\frac{y}{x} \cdot \frac{1}{y}} \]

   9. Applied egg-rr 27.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x} \cdot \frac{1}{y}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 56.8% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3e-18) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3e-18) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3d-18) then
        tmp = x
    else
        tmp = y / (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3e-18) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3e-18:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3e-18)
		tmp = x;
	else
		tmp = Float64(y / Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e-18)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3e-18], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.99999999999999983e-18

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 61.0%

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

   if 2.99999999999999983e-18 < y

   1. Initial program 99.5%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 99.5%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{\sin y}}} \]

      2. un-div-inv 99.5%

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

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around 0 29.0%

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

      1. *-commutative 29.0%

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

      2. clear-num 30.5%

         \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{y}{x}}} \]

      3. un-div-inv 30.5%

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

   9. Applied egg-rr 30.5%

      \[\leadsto \color{blue}{\frac{y}{\frac{y}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 56.4% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.5e-18) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-18) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.5d-18) then
        tmp = x
    else
        tmp = y * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-18) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.5e-18:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.5e-18)
		tmp = x;
	else
		tmp = Float64(y * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.5e-18)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.5e-18], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4999999999999999e-18

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 61.0%

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

   if 3.4999999999999999e-18 < y

   1. Initial program 99.5%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 99.5%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{\sin y}}} \]

      2. un-div-inv 99.5%

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

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around 0 29.0%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.2% accurate, 105.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.7%

   \[x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0 45.9%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))