Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.8% → 99.9%

Time: 10.4s

Alternatives: 10

Speedup: 1.0×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. associate-*l/ 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]
4. Add Preprocessing

5. Final simplification 99.9%

   \[\leadsto \frac{\sin x}{x} \cdot \sinh y \]
6. Add Preprocessing

Alternative 2: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 5e-18) (* (sin x) (/ y x)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 5e-18) {
		tmp = sin(x) * (y / x);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sinh(y) <= 5d-18) then
        tmp = sin(x) * (y / x)
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= 5e-18) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 5e-18:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 5e-18)
		tmp = Float64(sin(x) * Float64(y / x));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 5e-18)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 5e-18], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 5.00000000000000036e-18

   1. Initial program 81.3%

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

      1. *-commutative 81.3%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 51.2%

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

      1. associate-/l* 69.8%

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

      2. associate-/r/ 76.6%

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

   7. Simplified 76.6%

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

   if 5.00000000000000036e-18 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 76.1%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

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

Alternative 3: 56.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 5e-18)
   (/ y (* x (+ (* x 0.16666666666666666) (/ 1.0 x))))
   (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 5e-18) {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sinh(y) <= 5d-18) then
        tmp = y / (x * ((x * 0.16666666666666666d0) + (1.0d0 / x)))
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= 5e-18) {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 5e-18:
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 5e-18)
		tmp = Float64(y / Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x))));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 5e-18)
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 5e-18], N[(y / N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 5.00000000000000036e-18

   1. Initial program 81.3%

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

      1. *-commutative 81.3%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 51.2%

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

      1. associate-/l* 69.8%

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

   7. Simplified 69.8%

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

      1. clear-num 69.8%

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

      2. associate-/r/ 69.7%

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

   9. Applied egg-rr 69.7%

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

   10. Taylor expanded in x around 0 55.8%

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

   if 5.00000000000000036e-18 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 76.1%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. *-commutative 86.2%

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

   2. associate-*l/ 99.9%

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

   3. *-commutative 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Final simplification 99.9%

   \[\leadsto \sin x \cdot \frac{\sinh y}{x} \]
6. Add Preprocessing

Alternative 5: 66.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9200000:\\ \;\;\;\;\frac{\sin x}{-0.16666666666666666 \cdot \left(x \cdot y\right) + \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9200000.0)
   (/ (sin x) (+ (* -0.16666666666666666 (* x y)) (/ x y)))
   (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9200000.0) {
		tmp = sin(x) / ((-0.16666666666666666 * (x * y)) + (x / y));
	} else {
		tmp = sinh(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 <= 9200000.0d0) then
        tmp = sin(x) / (((-0.16666666666666666d0) * (x * y)) + (x / y))
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9200000.0) {
		tmp = Math.sin(x) / ((-0.16666666666666666 * (x * y)) + (x / y));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9200000.0:
		tmp = math.sin(x) / ((-0.16666666666666666 * (x * y)) + (x / y))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9200000.0)
		tmp = Float64(sin(x) / Float64(Float64(-0.16666666666666666 * Float64(x * y)) + Float64(x / y)));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9200000.0)
		tmp = sin(x) / ((-0.16666666666666666 * (x * y)) + (x / y));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9200000.0], N[(N[Sin[x], $MachinePrecision] / N[(N[(-0.16666666666666666 * N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.2e6

   1. Initial program 81.9%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{\sinh y}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 69.1%

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

   if 9.2e6 < y

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 77.0%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9200000:\\ \;\;\;\;\frac{\sin x}{-0.16666666666666666 \cdot \left(x \cdot y\right) + \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.1e-15) (/ y (/ x (sin x))) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-15) {
		tmp = y / (x / sin(x));
	} else {
		tmp = sinh(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 <= 1.1d-15) then
        tmp = y / (x / sin(x))
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-15) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.1e-15:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.1e-15)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.1e-15)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.1e-15], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.09999999999999993e-15

   1. Initial program 81.3%

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

      1. *-commutative 81.3%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 51.2%

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

      1. associate-/l* 69.8%

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

   7. Simplified 69.8%

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

   if 1.09999999999999993e-15 < y

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 76.1%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.3% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+27}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{x \cdot y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1e+27) y (/ 1.0 (/ x (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e+27) {
		tmp = y;
	} else {
		tmp = 1.0 / (x / (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 <= 1d+27) then
        tmp = y
    else
        tmp = 1.0d0 / (x / (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1e+27) {
		tmp = y;
	} else {
		tmp = 1.0 / (x / (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e+27:
		tmp = y
	else:
		tmp = 1.0 / (x / (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e+27)
		tmp = y;
	else
		tmp = Float64(1.0 / Float64(x / Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e+27)
		tmp = y;
	else
		tmp = 1.0 / (x / (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e+27], y, N[(1.0 / N[(x / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e27

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 50.3%

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

      1. associate-/l* 67.8%

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

   7. Simplified 67.8%

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

   8. Taylor expanded in x around 0 39.6%

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

   if 1e27 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 4.3%

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

      1. associate-/l* 4.3%

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

   7. Simplified 4.3%

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

      1. clear-num 4.3%

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

      2. inv-pow 4.3%

         \[\leadsto \color{blue}{{\left(\frac{\frac{x}{\sin x}}{y}\right)}^{-1}} \]

      3. associate-/l/ 4.3%

         \[\leadsto {\color{blue}{\left(\frac{x}{y \cdot \sin x}\right)}}^{-1} \]

   9. Applied egg-rr 4.3%

      \[\leadsto \color{blue}{{\left(\frac{x}{y \cdot \sin x}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 4.3%

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

   11. Simplified 4.3%

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

   12. Taylor expanded in x around 0 19.6%

       \[\leadsto \frac{1}{\frac{x}{\color{blue}{x \cdot y}}} \]
   13. Step-by-step derivation

       1. *-commutative 19.6%

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

   14. Simplified 19.6%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+27}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{x \cdot y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.2% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. *-commutative 86.2%

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

   2. associate-*l/ 99.9%

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

   3. *-commutative 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 40.4%

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

   1. associate-/l* 54.1%

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

   2. associate-/r/ 65.0%

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

7. Simplified 65.0%

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

   1. associate-*l/ 40.4%

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

   2. /-rgt-identity 40.4%

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

   3. associate-/r/ 40.3%

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

   4. associate-/l/ 54.0%

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

   5. associate-/r* 64.9%

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

9. Applied egg-rr 64.9%

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

10. Taylor expanded in x around 0 54.4%

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

11. Final simplification 54.4%

    \[\leadsto \frac{\frac{y}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x}} \]
12. Add Preprocessing

Alternative 9: 49.4% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{y}{x}}{\frac{1}{x}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (/ y x) (/ 1.0 x)))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (y / x) / (1.0 / x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. *-commutative 86.2%

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

   2. associate-*l/ 99.9%

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

   3. *-commutative 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 40.4%

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

   1. associate-/l* 54.1%

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

   2. associate-/r/ 65.0%

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

7. Simplified 65.0%

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

   1. associate-*l/ 40.4%

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

   2. /-rgt-identity 40.4%

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

   3. associate-/r/ 40.3%

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

   4. associate-/l/ 54.0%

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

   5. associate-/r* 64.9%

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

9. Applied egg-rr 64.9%

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

10. Taylor expanded in x around 0 53.9%

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

11. Final simplification 53.9%

    \[\leadsto \frac{\frac{y}{x}}{\frac{1}{x}} \]
12. Add Preprocessing

Alternative 10: 28.2% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 86.2%

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

   1. *-commutative 86.2%

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

   2. associate-*l/ 99.9%

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

   3. *-commutative 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 40.4%

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

   1. associate-/l* 54.1%

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

7. Simplified 54.1%

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

8. Taylor expanded in x around 0 32.0%

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

9. Final simplification 32.0%

   \[\leadsto y \]
10. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024019 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (* (sin x) (/ (sinh y) x))

  (/ (* (sin x) (sinh y)) x))