Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 5.8s

Alternatives: 7

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*r/ 89.5%

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

   2. associate-/l* 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 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 99.8%

   \[x \cdot \frac{\sin y}{y} \]

2. Final simplification 99.8%

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

Alternative 3: 64.6% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot \left(y \cdot 0.16666666666666666 + \frac{1}{y}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ x (* y (+ (* y 0.16666666666666666) (/ 1.0 y)))))

C

double code(double x, double y) {
	return x / (y * ((y * 0.16666666666666666) + (1.0 / y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (y * ((y * 0.16666666666666666d0) + (1.0d0 / y)))
end function

Java

public static double code(double x, double y) {
	return x / (y * ((y * 0.16666666666666666) + (1.0 / y)));
}

Python

def code(x, y):
	return x / (y * ((y * 0.16666666666666666) + (1.0 / y)))

Julia

function code(x, y)
	return Float64(x / Float64(y * Float64(Float64(y * 0.16666666666666666) + Float64(1.0 / y))))
end

MATLAB

function tmp = code(x, y)
	tmp = x / (y * ((y * 0.16666666666666666) + (1.0 / y)));
end

Wolfram

code[x_, y_] := N[(x / N[(y * N[(N[(y * 0.16666666666666666), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. clear-num 99.8%

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

   2. div-inv 99.8%

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

   3. div-inv 99.7%

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

   4. associate-/r* 87.9%

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

3. Applied egg-rr 87.9%

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

4. Taylor expanded in y around 0 55.6%

   \[\leadsto \frac{\frac{x}{y}}{\color{blue}{0.16666666666666666 \cdot y + \frac{1}{y}}} \]

5. Taylor expanded in x around 0 67.3%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(0.16666666666666666 \cdot y + \frac{1}{y}\right)}} \]

6. Final simplification 67.3%

   \[\leadsto \frac{x}{y \cdot \left(y \cdot 0.16666666666666666 + \frac{1}{y}\right)} \]

Alternative 4: 58.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{6}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.5) x (* (/ x y) (/ 6.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.5) {
		tmp = x;
	} else {
		tmp = (x / y) * (6.0 / 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 <= 2.5d0) then
        tmp = x
    else
        tmp = (x / y) * (6.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.5) {
		tmp = x;
	} else {
		tmp = (x / y) * (6.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.5:
		tmp = x
	else:
		tmp = (x / y) * (6.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.5)
		tmp = x;
	else
		tmp = Float64(Float64(x / y) * Float64(6.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.5)
		tmp = x;
	else
		tmp = (x / y) * (6.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.5], x, N[(N[(x / y), $MachinePrecision] * N[(6.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5

   1. Initial program 99.9%

      \[x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 70.5%

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

   if 2.5 < y

   1. Initial program 99.5%

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

      1. clear-num 99.5%

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

      2. div-inv 99.5%

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

      3. div-inv 99.5%

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

      4. associate-/r* 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in y around 0 37.7%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{0.16666666666666666 \cdot y + \frac{1}{y}}} \]

   5. Taylor expanded in y around inf 37.7%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{0.16666666666666666 \cdot y}} \]
   6. Step-by-step derivation

      1. *-commutative 37.7%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y \cdot 0.16666666666666666}} \]

   7. Simplified 37.7%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y \cdot 0.16666666666666666}} \]
   8. Step-by-step derivation

      1. div-inv 37.7%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{1}{y \cdot 0.16666666666666666}} \]

      2. *-commutative 37.7%

         \[\leadsto \frac{x}{y} \cdot \frac{1}{\color{blue}{0.16666666666666666 \cdot y}} \]

      3. associate-/r* 37.7%

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{y}} \]

      4. metadata-eval 37.7%

         \[\leadsto \frac{x}{y} \cdot \frac{\color{blue}{6}}{y} \]

   9. Applied egg-rr 37.7%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{6}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{6}{y}\\ \end{array} \]

Alternative 5: 57.5% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+54}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.5e+54) x (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.5e+54) {
		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 <= 2.5d+54) 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 <= 2.5e+54) {
		tmp = x;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.5e+54:
		tmp = x
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.5e+54)
		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 <= 2.5e+54)
		tmp = x;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.5e+54], x, N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.50000000000000003e54

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 67.1%

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

   if 2.50000000000000003e54 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 4.9%

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

      1. associate-*l/ 41.4%

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

   6. Applied egg-rr 41.4%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+54}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]

Alternative 6: 57.9% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 20000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 20000000.0) x (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 20000000.0) {
		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 <= 20000000.0d0) 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 <= 20000000.0) {
		tmp = x;
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 20000000.0:
		tmp = x
	else:
		tmp = y / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 20000000.0)
		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 <= 20000000.0)
		tmp = x;
	else
		tmp = y / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 20000000.0], x, N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e7

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 69.5%

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

   if 2e7 < y

   1. Initial program 99.6%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 5.2%

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

      1. associate-*l/ 36.8%

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

   6. Applied egg-rr 36.8%

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

      1. *-commutative 36.8%

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

      2. clear-num 38.4%

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

      3. un-div-inv 38.4%

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

   8. Applied egg-rr 38.4%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 20000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]

Alternative 7: 52.6% 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 99.8%

   \[x \cdot \frac{\sin y}{y} \]

2. Taylor expanded in y around 0 54.7%

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

3. Final simplification 54.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023302 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))