Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.8% → 99.8%

Time: 8.0s

Alternatives: 9

Speedup: 1.0×

• 50.1% 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: 99.8% 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 89.9%

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

   1. *-commutative 89.9%

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

   2. associate-/l* 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.002 \lor \neg \left(\sinh y \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) -0.002) (not (<= (sinh y) 5e-8)))
   (sinh y)
   (* (sin x) (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -0.002) || !(sinh(y) <= 5e-8)) {
		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) <= (-0.002d0)) .or. (.not. (sinh(y) <= 5d-8))) 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) <= -0.002) || !(Math.sinh(y) <= 5e-8)) {
		tmp = Math.sinh(y);
	} else {
		tmp = Math.sin(x) * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -0.002) or not (math.sinh(y) <= 5e-8):
		tmp = math.sinh(y)
	else:
		tmp = math.sin(x) * (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= -0.002) || !(sinh(y) <= 5e-8))
		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) <= -0.002) || ~((sinh(y) <= 5e-8)))
		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], -0.002], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 5e-8]], $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) < -2e-3 or 4.9999999999999998e-8 < (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 73.7%

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

   if -2e-3 < (sinh.f64 y) < 4.9999999999999998e-8

   1. Initial program 78.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 78.3%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.002 \lor \neg \left(\sinh y \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \]

Alternative 3: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.002 \lor \neg \left(\sinh y \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) -0.002) (not (<= (sinh y) 5e-8)))
   (sinh y)
   (* y (/ (sin x) x))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -0.002) || !(sinh(y) <= 5e-8)) {
		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) <= (-0.002d0)) .or. (.not. (sinh(y) <= 5d-8))) 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) <= -0.002) || !(Math.sinh(y) <= 5e-8)) {
		tmp = Math.sinh(y);
	} else {
		tmp = y * (Math.sin(x) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -0.002) or not (math.sinh(y) <= 5e-8):
		tmp = math.sinh(y)
	else:
		tmp = y * (math.sin(x) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= -0.002) || !(sinh(y) <= 5e-8))
		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) <= -0.002) || ~((sinh(y) <= 5e-8)))
		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], -0.002], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 5e-8]], $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) < -2e-3 or 4.9999999999999998e-8 < (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 73.7%

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

   if -2e-3 < (sinh.f64 y) < 4.9999999999999998e-8

   1. Initial program 78.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 78.3%

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

      1. associate-/l* 99.3%

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

   6. Simplified 99.3%

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

      1. clear-num 97.5%

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

      2. associate-/r/ 99.2%

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

      3. clear-num 99.2%

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

   8. Applied egg-rr 99.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.002 \lor \neg \left(\sinh y \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \end{array} \]

Alternative 4: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.002 \lor \neg \left(\sinh y \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) -0.002) (not (<= (sinh y) 5e-8)))
   (sinh y)
   (/ y (/ x (sin x)))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -0.002) || !(sinh(y) <= 5e-8)) {
		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) <= (-0.002d0)) .or. (.not. (sinh(y) <= 5d-8))) 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) <= -0.002) || !(Math.sinh(y) <= 5e-8)) {
		tmp = Math.sinh(y);
	} else {
		tmp = y / (x / Math.sin(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -0.002) or not (math.sinh(y) <= 5e-8):
		tmp = math.sinh(y)
	else:
		tmp = y / (x / math.sin(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= -0.002) || !(sinh(y) <= 5e-8))
		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) <= -0.002) || ~((sinh(y) <= 5e-8)))
		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], -0.002], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 5e-8]], $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) < -2e-3 or 4.9999999999999998e-8 < (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 73.7%

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

   if -2e-3 < (sinh.f64 y) < 4.9999999999999998e-8

   1. Initial program 78.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 78.3%

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

      1. associate-/l* 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.002 \lor \neg \left(\sinh y \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \end{array} \]

Alternative 5: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-61} \lor \neg \left(\sinh y \leq 5 \cdot 10^{-90}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) -2e-61) (not (<= (sinh y) 5e-90)))
   (sinh y)
   (/ x (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -2e-61) || !(sinh(y) <= 5e-90)) {
		tmp = sinh(y);
	} else {
		tmp = 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 ((sinh(y) <= (-2d-61)) .or. (.not. (sinh(y) <= 5d-90))) then
        tmp = sinh(y)
    else
        tmp = x / (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -2e-61) || !(Math.sinh(y) <= 5e-90)) {
		tmp = Math.sinh(y);
	} else {
		tmp = x / (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -2e-61) or not (math.sinh(y) <= 5e-90):
		tmp = math.sinh(y)
	else:
		tmp = x / (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= -2e-61) || !(sinh(y) <= 5e-90))
		tmp = sinh(y);
	else
		tmp = Float64(x / Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -2e-61) || ~((sinh(y) <= 5e-90)))
		tmp = sinh(y);
	else
		tmp = x / (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], -2e-61], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 5e-90]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -2.0000000000000001e-61 or 5.00000000000000019e-90 < (sinh.f64 y)

   1. Initial program 98.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 71.0%

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

   if -2.0000000000000001e-61 < (sinh.f64 y) < 5.00000000000000019e-90

   1. Initial program 75.3%

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

      1. associate-/l* 97.8%

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

   3. Simplified 97.8%

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

      1. *-un-lft-identity 97.8%

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

      2. div-inv 97.7%

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

      3. times-frac 75.3%

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

   5. Applied egg-rr 75.3%

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

   6. Taylor expanded in y around 0 75.3%

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

      1. *-commutative 75.3%

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

   8. Simplified 75.3%

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

   9. Taylor expanded in x around 0 22.2%

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

       1. *-commutative 22.2%

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

   11. Simplified 22.2%

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

       1. associate-*l/ 22.2%

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

       2. *-un-lft-identity 22.2%

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

       3. *-commutative 22.2%

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

       4. associate-/l* 79.6%

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

   13. Applied egg-rr 79.6%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-61} \lor \neg \left(\sinh y \leq 5 \cdot 10^{-90}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \]

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 89.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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: 50.8% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 39.9%

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

5. Taylor expanded in x around 0 22.3%

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

   1. *-un-lft-identity 22.3%

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

   2. times-frac 46.1%

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

   3. /-rgt-identity 46.1%

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

7. Applied egg-rr 46.1%

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

8. Final simplification 46.1%

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

Alternative 8: 50.4% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. associate-/l* 99.2%

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

3. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. div-inv 99.1%

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

   3. times-frac 89.8%

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

5. Applied egg-rr 89.8%

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

6. Taylor expanded in y around 0 39.9%

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

   1. *-commutative 39.9%

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

8. Simplified 39.9%

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

9. Taylor expanded in x around 0 22.2%

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

    1. *-commutative 22.2%

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

11. Simplified 22.2%

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

    1. associate-*l/ 22.3%

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

    2. *-un-lft-identity 22.3%

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

    3. *-commutative 22.3%

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

    4. associate-/l* 46.5%

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

13. Applied egg-rr 46.5%

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

14. Final simplification 46.5%

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

Alternative 9: 28.4% 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 89.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 39.9%

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

   1. associate-/l* 50.0%

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

6. Simplified 50.0%

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

7. Taylor expanded in x around 0 26.1%

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

8. Final simplification 26.1%

   \[\leadsto y \]

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 2023308 
(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))