Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.1% → 99.8%

Time: 8.6s

Alternatives: 8

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: 89.1% 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 86.1%

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

Alternative 2: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{-7}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{\sinh y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2e-7) (* y (/ (sin x) x)) (/ x (/ x (sinh y)))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2e-7) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = x / (x / 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) <= 2d-7) then
        tmp = y * (sin(x) / x)
    else
        tmp = x / (x / sinh(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= 2e-7) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = x / (x / Math.sinh(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 2e-7:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = x / (x / math.sinh(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 2e-7)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(x / Float64(x / sinh(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 2e-7)
		tmp = y * (sin(x) / x);
	else
		tmp = x / (x / sinh(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 2e-7], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(x / N[(x / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 1.9999999999999999e-7

   1. Initial program 81.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 52.2%

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

      1. associate-/l* 70.5%

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

   7. Simplified 70.5%

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

   if 1.9999999999999999e-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%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 78.7%

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

      1. clear-num 78.7%

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

      2. un-div-inv 78.7%

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

   7. Applied egg-rr 78.7%

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

Alternative 3: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{-7}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2e-7) (* y (/ (sin x) x)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2e-7) {
		tmp = y * (sin(x) / 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) <= 2d-7) then
        tmp = y * (sin(x) / 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) <= 2e-7) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 2e-7:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 2e-7)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 2e-7)
		tmp = y * (sin(x) / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 2e-7], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 1.9999999999999999e-7

   1. Initial program 81.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 52.2%

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

      1. associate-/l* 70.5%

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

   7. Simplified 70.5%

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

   if 1.9999999999999999e-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%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 78.7%

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

      1. clear-num 78.7%

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

      2. un-div-inv 78.7%

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

   7. Applied egg-rr 78.7%

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

      1. associate-/r/ 78.7%

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

      2. *-inverses 78.7%

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

      3. *-un-lft-identity 78.7%

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

      4. sinh-def 78.5%

         \[\leadsto \color{blue}{\frac{e^{y} - e^{-y}}{2}} \]

      5. div-sub 78.5%

         \[\leadsto \color{blue}{\frac{e^{y}}{2} - \frac{e^{-y}}{2}} \]

   9. Applied egg-rr 78.5%

      \[\leadsto \color{blue}{\frac{e^{y}}{2} - \frac{e^{-y}}{2}} \]
   10. Step-by-step derivation

       1. div-sub 78.5%

          \[\leadsto \color{blue}{\frac{e^{y} - e^{-y}}{2}} \]

       2. sinh-def 78.7%

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

   11. Simplified 78.7%

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

Alternative 4: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 10^{-33}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 1e-33) (/ x (/ x y)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 1e-33) {
		tmp = x / (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 (sinh(y) <= 1d-33) then
        tmp = x / (x / y)
    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) <= 1e-33) {
		tmp = x / (x / y);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 1e-33:
		tmp = x / (x / y)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 1e-33)
		tmp = Float64(x / Float64(x / y));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 1e-33)
		tmp = x / (x / y);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 1e-33], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 1.0000000000000001e-33

   1. Initial program 81.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 x around 0 76.5%

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

      1. clear-num 77.7%

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

      2. un-div-inv 77.8%

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

   7. Applied egg-rr 77.8%

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

   8. Taylor expanded in y around 0 64.6%

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

   if 1.0000000000000001e-33 < (sinh.f64 y)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 x around 0 76.0%

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

      1. clear-num 76.0%

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

      2. un-div-inv 76.0%

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

   7. Applied egg-rr 76.0%

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

      1. associate-/r/ 76.0%

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

      2. *-inverses 76.0%

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

      3. *-un-lft-identity 76.0%

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

      4. sinh-def 73.7%

         \[\leadsto \color{blue}{\frac{e^{y} - e^{-y}}{2}} \]

      5. div-sub 73.7%

         \[\leadsto \color{blue}{\frac{e^{y}}{2} - \frac{e^{-y}}{2}} \]

   9. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{\frac{e^{y}}{2} - \frac{e^{-y}}{2}} \]
   10. Step-by-step derivation

       1. div-sub 73.7%

          \[\leadsto \color{blue}{\frac{e^{y} - e^{-y}}{2}} \]

       2. sinh-def 76.0%

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

   11. Simplified 76.0%

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

Alternative 5: 74.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 x around 0 76.3%

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

Alternative 6: 53.2% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 550:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{1}{x} \cdot \frac{x}{\frac{y}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 550.0) (/ x (/ x y)) (/ x (* (/ 1.0 x) (/ x (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 550.0) {
		tmp = x / (x / y);
	} else {
		tmp = x / ((1.0 / x) * (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 <= 550.0d0) then
        tmp = x / (x / y)
    else
        tmp = x / ((1.0d0 / x) * (x / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 550.0) {
		tmp = x / (x / y);
	} else {
		tmp = x / ((1.0 / x) * (x / (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 550.0:
		tmp = x / (x / y)
	else:
		tmp = x / ((1.0 / x) * (x / (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 550.0)
		tmp = Float64(x / Float64(x / y));
	else
		tmp = Float64(x / Float64(Float64(1.0 / x) * Float64(x / Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 550.0)
		tmp = x / (x / y);
	else
		tmp = x / ((1.0 / x) * (x / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 550.0], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 / x), $MachinePrecision] * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 550

   1. Initial program 81.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 x around 0 75.7%

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

      1. clear-num 77.0%

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

      2. un-div-inv 77.1%

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

   7. Applied egg-rr 77.1%

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

   8. Taylor expanded in y around 0 64.0%

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

   if 550 < 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 x around 0 78.3%

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

      1. clear-num 78.3%

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

      2. un-div-inv 78.3%

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

   7. Applied egg-rr 78.3%

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

   8. Taylor expanded in y around 0 23.2%

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

      1. clear-num 29.4%

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

      2. lft-mult-inverse 29.4%

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

      3. associate-*l/ 29.4%

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

      4. *-un-lft-identity 29.4%

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

      5. add-sqr-sqrt 29.4%

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

      6. unpow2 29.4%

         \[\leadsto \frac{x}{\frac{\frac{x}{x}}{\frac{\color{blue}{{\left(\sqrt{y}\right)}^{2}}}{x}}} \]

      7. associate-/r* 29.4%

         \[\leadsto \frac{x}{\color{blue}{\frac{x}{x \cdot \frac{{\left(\sqrt{y}\right)}^{2}}{x}}}} \]

      8. *-un-lft-identity 29.4%

         \[\leadsto \frac{x}{\frac{\color{blue}{1 \cdot x}}{x \cdot \frac{{\left(\sqrt{y}\right)}^{2}}{x}}} \]

      9. times-frac 42.0%

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{x} \cdot \frac{x}{\frac{{\left(\sqrt{y}\right)}^{2}}{x}}}} \]

      10. unpow2 42.0%

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

      11. add-sqr-sqrt 42.0%

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

   10. Applied egg-rr 42.0%

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

Alternative 7: 50.4% 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 86.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 x around 0 76.3%

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

6. Taylor expanded in y around 0 55.2%

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

Alternative 8: 28.1% 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 86.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 41.0%

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

   1. associate-/l* 54.9%

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

7. Simplified 54.9%

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

8. Taylor expanded in x around 0 31.7%

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

9. Taylor expanded in y around 0 31.7%

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

Developer Target 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]

Reproduce

herbie shell --seed 2024116 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

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