Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.6% → 99.8%

Time: 7.9s

Alternatives: 9

Speedup: 1.0×

• 48.8% 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: 88.6% 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: 99.8% 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 92.2%

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

   1. associate-/l* 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. Add Preprocessing

Alternative 2: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8500000000:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8500000000.0) (/ y (/ x (sin x))) (log1p (expm1 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8500000000.0) {
		tmp = y / (x / sin(x));
	} else {
		tmp = log1p(expm1(y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8500000000.0) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.log1p(Math.expm1(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8500000000.0:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.log1p(math.expm1(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8500000000.0)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = log1p(expm1(y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8500000000.0], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[y] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5e9

   1. Initial program 89.8%

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

      1. associate-/l* 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 55.8%

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

      1. associate-/l* 66.0%

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

   7. Simplified 66.0%

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

      1. clear-num 66.0%

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

      2. un-div-inv 66.0%

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

   9. Applied egg-rr 66.0%

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

   if 8.5e9 < y

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 4.1%

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

      1. associate-/l* 4.1%

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

   7. Simplified 4.1%

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

      1. associate-*r/ 4.1%

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

      2. clear-num 4.1%

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

      3. *-commutative 4.1%

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

   9. Applied egg-rr 4.1%

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

   10. Taylor expanded in x around 0 3.6%

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

       1. remove-double-div 3.6%

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

       2. log1p-expm1-u 71.7%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]

   12. Applied egg-rr 71.7%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 56.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.5e-94) (/ (/ y x) (/ 1.0 x)) (* y (/ (sin x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.5e-94) {
		tmp = (y / x) / (1.0 / x);
	} else {
		tmp = y * (sin(x) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 4.5d-94) then
        tmp = (y / x) / (1.0d0 / x)
    else
        tmp = y * (sin(x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4.5e-94) {
		tmp = (y / x) / (1.0 / x);
	} else {
		tmp = y * (Math.sin(x) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4.5e-94:
		tmp = (y / x) / (1.0 / x)
	else:
		tmp = y * (math.sin(x) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.5e-94)
		tmp = Float64(Float64(y / x) / Float64(1.0 / x));
	else
		tmp = Float64(y * Float64(sin(x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4.5e-94)
		tmp = (y / x) / (1.0 / x);
	else
		tmp = y * (sin(x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.5e-94], N[(N[(y / x), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5000000000000002e-94

   1. Initial program 88.8%

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

      1. associate-/l* 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 35.4%

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

      1. *-commutative 35.4%

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

      2. associate-/l* 66.5%

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

   7. Applied egg-rr 66.5%

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

      1. associate-*r/ 35.4%

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

      2. clear-num 35.3%

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

      3. associate-/r* 46.5%

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

      4. clear-num 46.5%

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

      5. div-inv 46.5%

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

      6. associate-/r* 66.5%

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

   9. Applied egg-rr 66.5%

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

   10. Taylor expanded in x around 0 56.3%

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

   if 4.5000000000000002e-94 < x

   1. Initial program 98.8%

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

      1. associate-/l* 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 60.0%

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

      1. associate-/l* 61.0%

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

   7. Simplified 61.0%

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

Alternative 4: 63.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

   1. associate-/l* 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 43.7%

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

   1. *-commutative 43.7%

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

   2. associate-/l* 64.7%

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

7. Applied egg-rr 64.7%

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

Alternative 5: 31.0% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-10}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{x \cdot y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 4e-10) y (/ 1.0 (/ x (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4e-10) {
		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 (x <= 4d-10) 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 (x <= 4e-10) {
		tmp = y;
	} else {
		tmp = 1.0 / (x / (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4e-10:
		tmp = y
	else:
		tmp = 1.0 / (x / (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4e-10)
		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 (x <= 4e-10)
		tmp = y;
	else
		tmp = 1.0 / (x / (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4e-10], y, N[(1.0 / N[(x / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000015e-10

   1. Initial program 88.9%

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

      1. associate-/l* 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 37.9%

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

      1. associate-/l* 49.0%

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

   7. Simplified 49.0%

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

   8. Taylor expanded in x around 0 31.6%

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

   if 4.00000000000000015e-10 < x

   1. Initial program 99.8%

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

      1. associate-/l* 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 57.2%

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

      1. associate-/l* 57.2%

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

   7. Simplified 57.2%

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

      1. associate-*r/ 57.2%

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

      2. clear-num 56.3%

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

      3. *-commutative 56.3%

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

   9. Applied egg-rr 56.3%

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

   10. Taylor expanded in x around 0 15.0%

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

       1. *-commutative 15.0%

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

   12. Simplified 15.0%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-10}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{x \cdot y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 30.8% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{\frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3e-56) (/ 1.0 (/ 1.0 y)) (/ (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3e-56) {
		tmp = 1.0 / (1.0 / y);
	} else {
		tmp = (x * 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-56) then
        tmp = 1.0d0 / (1.0d0 / y)
    else
        tmp = (x * y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3e-56) {
		tmp = 1.0 / (1.0 / y);
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3e-56:
		tmp = 1.0 / (1.0 / y)
	else:
		tmp = (x * y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3e-56)
		tmp = Float64(1.0 / Float64(1.0 / y));
	else
		tmp = Float64(Float64(x * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e-56)
		tmp = 1.0 / (1.0 / y);
	else
		tmp = (x * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3e-56], N[(1.0 / N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.99999999999999989e-56

   1. Initial program 89.1%

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

      1. associate-/l* 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 54.0%

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

      1. associate-/l* 64.8%

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

   7. Simplified 64.8%

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

      1. associate-*r/ 54.0%

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

      2. clear-num 53.6%

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

      3. *-commutative 53.6%

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

   9. Applied egg-rr 53.6%

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

   10. Taylor expanded in x around 0 29.2%

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

   if 2.99999999999999989e-56 < y

   1. Initial program 99.9%

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

      1. associate-/l* 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 17.4%

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

   6. Taylor expanded in x around 0 16.7%

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

      1. *-commutative 16.7%

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

   8. Simplified 16.7%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{\frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.8% 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 92.2%

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

   1. associate-/l* 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 43.7%

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

   1. *-commutative 43.7%

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

   2. associate-/l* 64.7%

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

7. Applied egg-rr 64.7%

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

   1. associate-*r/ 43.7%

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

   2. clear-num 43.4%

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

   3. associate-/r* 51.2%

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

   4. clear-num 51.5%

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

   5. div-inv 51.4%

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

   6. associate-/r* 64.6%

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

9. Applied egg-rr 64.6%

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

10. Taylor expanded in x around 0 48.8%

    \[\leadsto \frac{\frac{y}{x}}{\color{blue}{\frac{1}{x}}} \]
11. Add Preprocessing

Alternative 8: 28.1% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

   1. associate-/l* 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 43.7%

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

   1. associate-/l* 51.5%

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

7. Simplified 51.5%

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

   1. associate-*r/ 43.7%

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

   2. clear-num 43.4%

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

   3. *-commutative 43.4%

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

9. Applied egg-rr 43.4%

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

10. Taylor expanded in x around 0 23.9%

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{y}}} \]
11. Add Preprocessing

Alternative 9: 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 92.2%

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

   1. associate-/l* 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 43.7%

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

   1. associate-/l* 51.5%

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

7. Simplified 51.5%

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

8. Taylor expanded in x around 0 23.7%

   \[\leadsto \color{blue}{y} \]
9. Add Preprocessing

Developer target: 99.8% 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 2024089 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

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

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