Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.2% → 99.9%

Time: 7.6s

Alternatives: 9

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: 89.2% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

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

   1. *-commutative 91.8%

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

   2. associate-/l* 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-8} \lor \neg \left(\sinh y \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) -5e-8) (not (<= (sinh y) 5e-7)))
   (sinh y)
   (* (sin x) (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -5e-8) || !(sinh(y) <= 5e-7)) {
		tmp = sinh(y);
	} else {
		tmp = sin(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 ((sinh(y) <= (-5d-8)) .or. (.not. (sinh(y) <= 5d-7))) then
        tmp = sinh(y)
    else
        tmp = sin(x) * (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -5e-8) || !(Math.sinh(y) <= 5e-7)) {
		tmp = Math.sinh(y);
	} else {
		tmp = Math.sin(x) * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -5e-8) or not (math.sinh(y) <= 5e-7):
		tmp = math.sinh(y)
	else:
		tmp = math.sin(x) * (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= -5e-8) || !(sinh(y) <= 5e-7))
		tmp = sinh(y);
	else
		tmp = Float64(sin(x) * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -5e-8) || ~((sinh(y) <= 5e-7)))
		tmp = sinh(y);
	else
		tmp = sin(x) * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], -5e-8], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 5e-7]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -4.9999999999999998e-8 or 4.99999999999999977e-7 < (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. Taylor expanded in x around 0 75.2%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]

   if -4.9999999999999998e-8 < (sinh.f64 y) < 4.99999999999999977e-7

   1. Initial program 81.1%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 81.0%

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

      1. associate-/l* 99.8%

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

      2. associate-/r/ 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-8} \lor \neg \left(\sinh y \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \]

Alternative 3: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-8} \lor \neg \left(\sinh y \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) -5e-8) (not (<= (sinh y) 5e-7)))
   (sinh y)
   (* y (/ (sin x) x))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -5e-8) || !(sinh(y) <= 5e-7)) {
		tmp = sinh(y);
	} 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 ((sinh(y) <= (-5d-8)) .or. (.not. (sinh(y) <= 5d-7))) then
        tmp = sinh(y)
    else
        tmp = y * (sin(x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -5e-8) || !(Math.sinh(y) <= 5e-7)) {
		tmp = Math.sinh(y);
	} else {
		tmp = y * (Math.sin(x) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -5e-8) or not (math.sinh(y) <= 5e-7):
		tmp = math.sinh(y)
	else:
		tmp = y * (math.sin(x) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= -5e-8) || !(sinh(y) <= 5e-7))
		tmp = sinh(y);
	else
		tmp = Float64(y * Float64(sin(x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -5e-8) || ~((sinh(y) <= 5e-7)))
		tmp = sinh(y);
	else
		tmp = y * (sin(x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], -5e-8], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 5e-7]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -4.9999999999999998e-8 or 4.99999999999999977e-7 < (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. Taylor expanded in x around 0 75.2%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]

   if -4.9999999999999998e-8 < (sinh.f64 y) < 4.99999999999999977e-7

   1. Initial program 81.1%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 81.0%

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

      1. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

      1. associate-/l* 81.0%

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

      2. *-commutative 81.0%

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

      3. associate-*l/ 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-8} \lor \neg \left(\sinh y \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \end{array} \]

Alternative 4: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-8} \lor \neg \left(\sinh y \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) -5e-8) (not (<= (sinh y) 5e-7)))
   (sinh y)
   (/ y (/ x (sin x)))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -5e-8) || !(sinh(y) <= 5e-7)) {
		tmp = sinh(y);
	} else {
		tmp = y / (x / sin(x));
	}
	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-8)) .or. (.not. (sinh(y) <= 5d-7))) then
        tmp = sinh(y)
    else
        tmp = y / (x / sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -5e-8) || !(Math.sinh(y) <= 5e-7)) {
		tmp = Math.sinh(y);
	} else {
		tmp = y / (x / Math.sin(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -5e-8) or not (math.sinh(y) <= 5e-7):
		tmp = math.sinh(y)
	else:
		tmp = y / (x / math.sin(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= -5e-8) || !(sinh(y) <= 5e-7))
		tmp = sinh(y);
	else
		tmp = Float64(y / Float64(x / sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -5e-8) || ~((sinh(y) <= 5e-7)))
		tmp = sinh(y);
	else
		tmp = y / (x / sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], -5e-8], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 5e-7]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -4.9999999999999998e-8 or 4.99999999999999977e-7 < (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. Taylor expanded in x around 0 75.2%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]

   if -4.9999999999999998e-8 < (sinh.f64 y) < 4.99999999999999977e-7

   1. Initial program 81.1%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 81.0%

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

      1. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-8} \lor \neg \left(\sinh y \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \end{array} \]

Alternative 5: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-8} \lor \neg \left(\sinh y \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) -5e-8) (not (<= (sinh y) 5e-7)))
   (sinh y)
   (/ (/ y x) (+ (* x 0.16666666666666666) (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -5e-8) || !(sinh(y) <= 5e-7)) {
		tmp = sinh(y);
	} else {
		tmp = (y / x) / ((x * 0.16666666666666666) + (1.0 / x));
	}
	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-8)) .or. (.not. (sinh(y) <= 5d-7))) then
        tmp = sinh(y)
    else
        tmp = (y / x) / ((x * 0.16666666666666666d0) + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -5e-8) || !(Math.sinh(y) <= 5e-7)) {
		tmp = Math.sinh(y);
	} else {
		tmp = (y / x) / ((x * 0.16666666666666666) + (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -5e-8) or not (math.sinh(y) <= 5e-7):
		tmp = math.sinh(y)
	else:
		tmp = (y / x) / ((x * 0.16666666666666666) + (1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= -5e-8) || !(sinh(y) <= 5e-7))
		tmp = sinh(y);
	else
		tmp = Float64(Float64(y / x) / Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -5e-8) || ~((sinh(y) <= 5e-7)))
		tmp = sinh(y);
	else
		tmp = (y / x) / ((x * 0.16666666666666666) + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], -5e-8], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 5e-7]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(N[(y / x), $MachinePrecision] / N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -4.9999999999999998e-8 or 4.99999999999999977e-7 < (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. Taylor expanded in x around 0 75.2%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]

   if -4.9999999999999998e-8 < (sinh.f64 y) < 4.99999999999999977e-7

   1. Initial program 81.1%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 81.0%

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

      1. associate-/l* 99.8%

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

      2. associate-/r/ 99.7%

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

   6. Simplified 99.7%

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

      1. associate-/r/ 99.8%

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

      2. div-inv 99.6%

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

      3. associate-/r* 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in x around 0 72.0%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{0.16666666666666666 \cdot x + \frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-8} \lor \neg \left(\sinh y \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x}}\\ \end{array} \]

Alternative 6: 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 91.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. Final simplification 99.9%

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

Alternative 7: 51.0% 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 91.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. Taylor expanded in y around 0 38.3%

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

   1. associate-/l* 46.5%

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

   2. associate-/r/ 57.4%

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

6. Simplified 57.4%

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

   1. associate-/r/ 46.5%

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

   2. div-inv 46.4%

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

   3. associate-/r* 57.3%

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

8. Applied egg-rr 57.3%

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

9. Taylor expanded in x around 0 44.9%

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

10. Final simplification 44.9%

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

Alternative 8: 50.1% 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 91.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. Taylor expanded in y around 0 38.3%

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

   1. associate-/l* 46.5%

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

   2. associate-/r/ 57.4%

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

6. Simplified 57.4%

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

   1. associate-/r/ 46.5%

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

   2. div-inv 46.4%

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

   3. associate-/r* 57.3%

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

8. Applied egg-rr 57.3%

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

9. Taylor expanded in x around 0 44.5%

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

10. Final simplification 44.5%

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

Alternative 9: 28.0% 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 91.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. Taylor expanded in y around 0 38.3%

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

   1. associate-/l* 46.5%

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

   2. associate-/r/ 57.4%

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

6. Simplified 57.4%

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

7. Taylor expanded in x around 0 25.6%

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

8. Final simplification 25.6%

   \[\leadsto y \]

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 2023320 
(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))