Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.3% → 99.9%

Time: 8.4s

Alternatives: 13

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: 89.3% 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 87.7%

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

   1. associate-*r/ 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. Step-by-step derivation

   1. add-log-exp 65.4%

      \[\leadsto \color{blue}{\log \left(e^{\sin x \cdot \frac{\sinh y}{x}}\right)} \]

   2. *-un-lft-identity 65.4%

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

   3. log-prod 65.4%

      \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sin x \cdot \frac{\sinh y}{x}}\right)} \]

   4. metadata-eval 65.4%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{x}}\right) \]

   5. add-log-exp 99.9%

      \[\leadsto 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{x}} \]
6. Step-by-step derivation

   1. +-lft-identity 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-/r/ 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-6}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) (- INFINITY))
   (sinh y)
   (if (<= (sinh y) 1e-6) (* (sin x) (/ y x)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -((double) INFINITY)) {
		tmp = sinh(y);
	} else if (sinh(y) <= 1e-6) {
		tmp = sin(x) * (y / x);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 1e-6) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -math.inf:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 1e-6:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= Float64(-Inf))
		tmp = sinh(y);
	elseif (sinh(y) <= 1e-6)
		tmp = Float64(sin(x) * Float64(y / x));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= -Inf)
		tmp = sinh(y);
	elseif (sinh(y) <= 1e-6)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 1e-6], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -inf.0 or 9.99999999999999955e-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 77.9%

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

   if -inf.0 < (sinh.f64 y) < 9.99999999999999955e-7

   1. Initial program 72.9%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 72.8%

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

      1. associate-/l* 99.7%

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

      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 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-6}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 3: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-6}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) (- INFINITY))
   (sinh y)
   (if (<= (sinh y) 1e-6) (* y (/ (sin x) x)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -((double) INFINITY)) {
		tmp = sinh(y);
	} else if (sinh(y) <= 1e-6) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 1e-6) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -math.inf:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 1e-6:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= Float64(-Inf))
		tmp = sinh(y);
	elseif (sinh(y) <= 1e-6)
		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) <= -Inf)
		tmp = sinh(y);
	elseif (sinh(y) <= 1e-6)
		tmp = y * (sin(x) / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 1e-6], 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) < -inf.0 or 9.99999999999999955e-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 77.9%

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

   if -inf.0 < (sinh.f64 y) < 9.99999999999999955e-7

   1. Initial program 72.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.7%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-6}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 4: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-6}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) (- INFINITY))
   (sinh y)
   (if (<= (sinh y) 1e-6) (/ y (/ x (sin x))) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -((double) INFINITY)) {
		tmp = sinh(y);
	} else if (sinh(y) <= 1e-6) {
		tmp = y / (x / sin(x));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 1e-6) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -math.inf:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 1e-6:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= Float64(-Inf))
		tmp = sinh(y);
	elseif (sinh(y) <= 1e-6)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= -Inf)
		tmp = sinh(y);
	elseif (sinh(y) <= 1e-6)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 1e-6], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -inf.0 or 9.99999999999999955e-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 77.9%

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

   if -inf.0 < (sinh.f64 y) < 9.99999999999999955e-7

   1. Initial program 72.9%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 72.8%

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

      1. associate-/l* 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-6}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 5: 72.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-24}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-156}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -2e-24)
   (sinh y)
   (if (<= (sinh y) 1e-156) (/ x (/ x y)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -2e-24) {
		tmp = sinh(y);
	} else if (sinh(y) <= 1e-156) {
		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) <= (-2d-24)) then
        tmp = sinh(y)
    else if (sinh(y) <= 1d-156) 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) <= -2e-24) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 1e-156) {
		tmp = x / (x / y);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -2e-24:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 1e-156:
		tmp = x / (x / y)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -2e-24)
		tmp = sinh(y);
	elseif (sinh(y) <= 1e-156)
		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) <= -2e-24)
		tmp = sinh(y);
	elseif (sinh(y) <= 1e-156)
		tmp = x / (x / y);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -2e-24], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 1e-156], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -1.99999999999999985e-24 or 1.00000000000000004e-156 < (sinh.f64 y)

   1. Initial program 98.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 x around 0 73.0%

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

   if -1.99999999999999985e-24 < (sinh.f64 y) < 1.00000000000000004e-156

   1. Initial program 65.4%

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

   2. Taylor expanded in y around 0 65.4%

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

   3. Taylor expanded in x around 0 16.4%

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

      1. associate-/l* 50.8%

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

      2. associate-/r/ 72.6%

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

   5. Applied egg-rr 72.6%

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

      1. frac-2neg 72.6%

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

      2. clear-num 75.7%

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

      3. associate-*l/ 75.8%

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

      4. *-un-lft-identity 75.8%

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

      5. frac-2neg 75.8%

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

   7. Applied egg-rr 75.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-24}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-156}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. associate-*r/ 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. Final simplification 99.9%

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

Alternative 7: 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 87.7%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 8: 87.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \sinh y\\ \mathbf{if}\;y \leq -0.0017:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+75}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ 1.0 (* (* x x) -0.16666666666666666)) (sinh y))))
   (if (<= y -0.0017)
     t_0
     (if (<= y 1.7e-6) (/ y (/ x (sin x))) (if (<= y 9e+75) (sinh y) t_0)))))

C

double code(double x, double y) {
	double t_0 = (1.0 + ((x * x) * -0.16666666666666666)) * sinh(y);
	double tmp;
	if (y <= -0.0017) {
		tmp = t_0;
	} else if (y <= 1.7e-6) {
		tmp = y / (x / sin(x));
	} else if (y <= 9e+75) {
		tmp = sinh(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + ((x * x) * (-0.16666666666666666d0))) * sinh(y)
    if (y <= (-0.0017d0)) then
        tmp = t_0
    else if (y <= 1.7d-6) then
        tmp = y / (x / sin(x))
    else if (y <= 9d+75) then
        tmp = sinh(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 + ((x * x) * -0.16666666666666666)) * Math.sinh(y);
	double tmp;
	if (y <= -0.0017) {
		tmp = t_0;
	} else if (y <= 1.7e-6) {
		tmp = y / (x / Math.sin(x));
	} else if (y <= 9e+75) {
		tmp = Math.sinh(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 + ((x * x) * -0.16666666666666666)) * math.sinh(y)
	tmp = 0
	if y <= -0.0017:
		tmp = t_0
	elif y <= 1.7e-6:
		tmp = y / (x / math.sin(x))
	elif y <= 9e+75:
		tmp = math.sinh(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)) * sinh(y))
	tmp = 0.0
	if (y <= -0.0017)
		tmp = t_0;
	elseif (y <= 1.7e-6)
		tmp = Float64(y / Float64(x / sin(x)));
	elseif (y <= 9e+75)
		tmp = sinh(y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 + ((x * x) * -0.16666666666666666)) * sinh(y);
	tmp = 0.0;
	if (y <= -0.0017)
		tmp = t_0;
	elseif (y <= 1.7e-6)
		tmp = y / (x / sin(x));
	elseif (y <= 9e+75)
		tmp = sinh(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0017], t$95$0, If[LessEqual[y, 1.7e-6], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e+75], N[Sinh[y], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.00169999999999999991 or 9.0000000000000007e75 < 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 85.7%

      \[\leadsto \color{blue}{\left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} \cdot \sinh y \]
   5. Step-by-step derivation

      1. *-commutative 85.7%

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right) \cdot \sinh y \]

      2. unpow2 85.7%

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

   6. Simplified 85.7%

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

   if -0.00169999999999999991 < y < 1.70000000000000003e-6

   1. Initial program 72.9%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 72.8%

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

      1. associate-/l* 99.7%

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

   6. Simplified 99.7%

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

   if 1.70000000000000003e-6 < y < 9.0000000000000007e75

   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 95.2%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0017:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \sinh y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+75}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \sinh y\\ \end{array} \]

Alternative 9: 53.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.9e+138) (* x (/ y x)) (sqrt (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.9e+138) {
		tmp = x * (y / x);
	} else {
		tmp = sqrt((y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.9d+138) then
        tmp = x * (y / x)
    else
        tmp = sqrt((y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.9e+138) {
		tmp = x * (y / x);
	} else {
		tmp = Math.sqrt((y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.9e+138:
		tmp = x * (y / x)
	else:
		tmp = math.sqrt((y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.9e+138)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = sqrt(Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.9e+138)
		tmp = x * (y / x);
	else
		tmp = sqrt((y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.9e+138], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9000000000000001e138

   1. Initial program 85.5%

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

   2. Taylor expanded in y around 0 41.4%

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

   3. Taylor expanded in x around 0 18.6%

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

      1. associate-/l* 29.9%

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

      2. associate-/r/ 51.1%

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

   5. Applied egg-rr 51.1%

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

   if 2.9000000000000001e138 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 5.3%

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

   3. Taylor expanded in x around 0 19.3%

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

      1. div-inv 19.3%

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

      2. associate-*l* 4.9%

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

      3. div-inv 4.9%

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

      4. *-inverses 4.9%

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

      5. *-commutative 4.9%

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

      6. *-un-lft-identity 4.9%

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

      7. add-sqr-sqrt 4.9%

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

      8. sqrt-unprod 69.5%

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

   5. Applied egg-rr 69.5%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \]

Alternative 10: 47.6% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+177}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+272}:\\ \;\;\;\;\frac{x \cdot \left(-y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.8e+177)
   (* x (/ y x))
   (if (<= x 5.3e+272) (/ (* x (- y)) x) (/ (* y x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.8e+177) {
		tmp = x * (y / x);
	} else if (x <= 5.3e+272) {
		tmp = (x * -y) / x;
	} else {
		tmp = (y * 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 <= 5.8d+177) then
        tmp = x * (y / x)
    else if (x <= 5.3d+272) then
        tmp = (x * -y) / x
    else
        tmp = (y * x) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.8e+177) {
		tmp = x * (y / x);
	} else if (x <= 5.3e+272) {
		tmp = (x * -y) / x;
	} else {
		tmp = (y * x) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.8e+177:
		tmp = x * (y / x)
	elif x <= 5.3e+272:
		tmp = (x * -y) / x
	else:
		tmp = (y * x) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.8e+177)
		tmp = Float64(x * Float64(y / x));
	elseif (x <= 5.3e+272)
		tmp = Float64(Float64(x * Float64(-y)) / x);
	else
		tmp = Float64(Float64(y * x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.8e+177)
		tmp = x * (y / x);
	elseif (x <= 5.3e+272)
		tmp = (x * -y) / x;
	else
		tmp = (y * x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.8e+177], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.3e+272], N[(N[(x * (-y)), $MachinePrecision] / x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.80000000000000027e177

   1. Initial program 86.2%

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

   2. Taylor expanded in y around 0 34.6%

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

   3. Taylor expanded in x around 0 19.0%

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

      1. associate-/l* 29.1%

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

      2. associate-/r/ 52.5%

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

   5. Applied egg-rr 52.5%

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

   if 5.80000000000000027e177 < x < 5.30000000000000011e272

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 43.1%

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

   3. Taylor expanded in x around 0 10.4%

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

      1. associate-/l* 2.6%

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

      2. associate-/r/ 18.4%

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

   5. Applied egg-rr 18.4%

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

      1. associate-*l/ 10.4%

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

      2. frac-2neg 10.4%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 21.6%

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

      5. sqr-neg 21.6%

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

      6. sqrt-unprod 39.5%

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

      7. add-sqr-sqrt 39.5%

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

   7. Applied egg-rr 39.5%

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

   if 5.30000000000000011e272 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 57.8%

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

   3. Taylor expanded in x around 0 42.2%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+177}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+272}:\\ \;\;\;\;\frac{x \cdot \left(-y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \]

Alternative 11: 51.1% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 280:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(-y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 280.0) (* x (/ y x)) (/ (* y (- y)) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 280.0) {
		tmp = x * (y / x);
	} else {
		tmp = (y * -y) / 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 <= 280.0d0) then
        tmp = x * (y / x)
    else
        tmp = (y * -y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 280.0) {
		tmp = x * (y / x);
	} else {
		tmp = (y * -y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 280.0:
		tmp = x * (y / x)
	else:
		tmp = (y * -y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 280.0)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(Float64(y * Float64(-y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 280.0)
		tmp = x * (y / x);
	else
		tmp = (y * -y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 280.0], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(y * (-y)), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 280

   1. Initial program 84.0%

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

   2. Taylor expanded in y around 0 31.3%

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

   3. Taylor expanded in x around 0 20.1%

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

      1. associate-/l* 33.2%

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

      2. associate-/r/ 59.1%

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

   5. Applied egg-rr 59.1%

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

   if 280 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 50.5%

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

   3. Taylor expanded in x around 0 14.1%

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

      1. clear-num 14.1%

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

      2. inv-pow 14.1%

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

      3. *-commutative 14.1%

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

      4. associate-/r* 3.4%

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

      5. *-inverses 3.4%

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

   5. Applied egg-rr 3.4%

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

      1. unpow-1 3.4%

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

      2. remove-double-div 3.4%

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

      3. add-sqr-sqrt 1.7%

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

      4. sqrt-unprod 26.5%

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

      5. sqr-neg 26.5%

         \[\leadsto \sqrt{\color{blue}{\left(-y\right) \cdot \left(-y\right)}} \]

      6. sqrt-unprod 2.1%

         \[\leadsto \color{blue}{\sqrt{-y} \cdot \sqrt{-y}} \]

      7. add-sqr-sqrt 4.4%

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

      8. neg-sub0 4.4%

         \[\leadsto \color{blue}{0 - y} \]

      9. metadata-eval 4.4%

         \[\leadsto \color{blue}{\log 1} - y \]

      10. flip-- 31.5%

          \[\leadsto \color{blue}{\frac{\log 1 \cdot \log 1 - y \cdot y}{\log 1 + y}} \]

      11. metadata-eval 31.5%

          \[\leadsto \frac{\color{blue}{0} \cdot \log 1 - y \cdot y}{\log 1 + y} \]

      12. metadata-eval 31.5%

          \[\leadsto \frac{0 \cdot \color{blue}{0} - y \cdot y}{\log 1 + y} \]

      13. metadata-eval 31.5%

          \[\leadsto \frac{\color{blue}{0} - y \cdot y}{\log 1 + y} \]

      14. add-log-exp 0.9%

          \[\leadsto \frac{0 - y \cdot y}{\log 1 + \color{blue}{\log \left(e^{y}\right)}} \]

      15. log-prod 0.9%

          \[\leadsto \frac{0 - y \cdot y}{\color{blue}{\log \left(1 \cdot e^{y}\right)}} \]

      16. *-un-lft-identity 0.9%

          \[\leadsto \frac{0 - y \cdot y}{\log \color{blue}{\left(e^{y}\right)}} \]

      17. add-log-exp 31.5%

          \[\leadsto \frac{0 - y \cdot y}{\color{blue}{y}} \]

   7. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{\frac{0 - y \cdot y}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 280:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(-y\right)}{y}\\ \end{array} \]

Alternative 12: 49.3% 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 87.7%

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

2. Taylor expanded in y around 0 35.9%

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

3. Taylor expanded in x around 0 18.7%

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

   1. associate-/l* 26.1%

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

   2. associate-/r/ 49.1%

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

5. Applied egg-rr 49.1%

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

6. Final simplification 49.1%

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

Alternative 13: 27.1% 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 87.7%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 48.1%

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

5. Taylor expanded in x around 0 26.1%

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

6. Final simplification 26.1%

   \[\leadsto y \]

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 2023229 
(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))