Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.0% → 99.9%

Time: 7.9s

Alternatives: 9

Speedup: 1.0×

• 48.7% 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: 89.0% 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

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

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 83.6%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 2: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 1.5 \cdot 10^{-22}\right):\\ \;\;\;\;\sinh y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) (- INFINITY)) (not (<= (sinh y) 1.5e-22)))
   (* (sinh y) (+ 1.0 (* (* x x) -0.16666666666666666)))
   (* (/ (sin x) x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -((double) INFINITY)) || !(sinh(y) <= 1.5e-22)) {
		tmp = sinh(y) * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = (sin(x) / x) * y;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -Double.POSITIVE_INFINITY) || !(Math.sinh(y) <= 1.5e-22)) {
		tmp = Math.sinh(y) * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = (Math.sin(x) / x) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -math.inf) or not (math.sinh(y) <= 1.5e-22):
		tmp = math.sinh(y) * (1.0 + ((x * x) * -0.16666666666666666))
	else:
		tmp = (math.sin(x) / x) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= Float64(-Inf)) || !(sinh(y) <= 1.5e-22))
		tmp = Float64(sinh(y) * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(sin(x) / x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -Inf) || ~((sinh(y) <= 1.5e-22)))
		tmp = sinh(y) * (1.0 + ((x * x) * -0.16666666666666666));
	else
		tmp = (sin(x) / x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 1.5e-22]], $MachinePrecision]], N[(N[Sinh[y], $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -inf.0 or 1.5e-22 < (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 77.6%

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

      1. *-commutative 77.6%

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

      2. unpow2 77.6%

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

   6. Simplified 77.6%

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

   if -inf.0 < (sinh.f64 y) < 1.5e-22

   1. Initial program 67.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 1.5 \cdot 10^{-22}\right):\\ \;\;\;\;\sinh y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \end{array} \]

Alternative 3: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 1.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) (- INFINITY))
   (sinh y)
   (if (<= (sinh y) 1.5e-22) (* (/ (sin x) x) y) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -((double) INFINITY)) {
		tmp = sinh(y);
	} else if (sinh(y) <= 1.5e-22) {
		tmp = (sin(x) / x) * y;
	} 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) <= 1.5e-22) {
		tmp = (Math.sin(x) / x) * y;
	} 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) <= 1.5e-22:
		tmp = (math.sin(x) / x) * y
	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) <= 1.5e-22)
		tmp = Float64(Float64(sin(x) / x) * y);
	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) <= 1.5e-22)
		tmp = (sin(x) / x) * y;
	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], 1.5e-22], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -inf.0 or 1.5e-22 < (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 73.6%

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

   if -inf.0 < (sinh.f64 y) < 1.5e-22

   1. Initial program 67.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 1.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 4: 74.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -1 \cdot 10^{-89}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 1.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -1e-89)
   (sinh y)
   (if (<= (sinh y) 1.5e-22) (/ x (/ x y)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -1e-89) {
		tmp = sinh(y);
	} else if (sinh(y) <= 1.5e-22) {
		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-89)) then
        tmp = sinh(y)
    else if (sinh(y) <= 1.5d-22) 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-89) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 1.5e-22) {
		tmp = x / (x / y);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -1e-89:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 1.5e-22:
		tmp = x / (x / y)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -1e-89)
		tmp = sinh(y);
	elseif (sinh(y) <= 1.5e-22)
		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-89)
		tmp = sinh(y);
	elseif (sinh(y) <= 1.5e-22)
		tmp = x / (x / y);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -1e-89], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 1.5e-22], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -1.00000000000000004e-89 or 1.5e-22 < (sinh.f64 y)

   1. Initial program 98.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 71.2%

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

   if -1.00000000000000004e-89 < (sinh.f64 y) < 1.5e-22

   1. Initial program 62.6%

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

      1. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 62.6%

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

      1. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

      1. associate-/l* 62.6%

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

      2. clear-num 61.5%

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

      3. associate-/r/ 62.6%

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

      4. *-commutative 62.6%

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

   8. Applied egg-rr 62.6%

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

   9. Taylor expanded in x around 0 13.6%

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

       1. associate-*l/ 13.6%

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

       2. *-un-lft-identity 13.6%

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

       3. *-commutative 13.6%

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

       4. associate-/l* 73.9%

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

   11. Applied egg-rr 73.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -1 \cdot 10^{-89}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 1.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 5: 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 83.6%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 6: 51.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.15e-23) (* x (/ y x)) (sqrt (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.15e-23) {
		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 (x <= 1.15d-23) 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 (x <= 1.15e-23) {
		tmp = x * (y / x);
	} else {
		tmp = Math.sqrt((y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.15e-23:
		tmp = x * (y / x)
	else:
		tmp = math.sqrt((y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.15e-23)
		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 (x <= 1.15e-23)
		tmp = x * (y / x);
	else
		tmp = sqrt((y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.15e-23], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15000000000000005e-23

   1. Initial program 77.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. Taylor expanded in y around 0 33.3%

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

   5. Taylor expanded in x around 0 19.5%

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

      1. associate-/l* 38.8%

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

      2. associate-/r/ 58.6%

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

   7. Applied egg-rr 58.6%

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

   if 1.15000000000000005e-23 < x

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. Taylor expanded in y around 0 51.5%

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

   5. Taylor expanded in x around 0 9.5%

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

      1. associate-/l* 5.4%

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

      2. *-inverses 5.4%

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

      3. /-rgt-identity 5.4%

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

      4. add-sqr-sqrt 2.9%

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

      5. sqrt-unprod 31.5%

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

   7. Applied egg-rr 31.5%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \]

Alternative 7: 50.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y} + y \cdot -0.16666666666666666\\ \mathbf{if}\;x \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{t_0} + -0.16666666666666666 \cdot \frac{x \cdot x}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 y) (* y -0.16666666666666666))))
   (if (<= x 6.8e+153)
     (* x (/ y x))
     (if (<= x 6.6e+180)
       (+ (/ 1.0 t_0) (* -0.16666666666666666 (/ (* x x) t_0)))
       (/ x (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (1.0 / y) + (y * -0.16666666666666666);
	double tmp;
	if (x <= 6.8e+153) {
		tmp = x * (y / x);
	} else if (x <= 6.6e+180) {
		tmp = (1.0 / t_0) + (-0.16666666666666666 * ((x * x) / t_0));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / y) + (y * (-0.16666666666666666d0))
    if (x <= 6.8d+153) then
        tmp = x * (y / x)
    else if (x <= 6.6d+180) then
        tmp = (1.0d0 / t_0) + ((-0.16666666666666666d0) * ((x * x) / t_0))
    else
        tmp = x / (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / y) + (y * -0.16666666666666666);
	double tmp;
	if (x <= 6.8e+153) {
		tmp = x * (y / x);
	} else if (x <= 6.6e+180) {
		tmp = (1.0 / t_0) + (-0.16666666666666666 * ((x * x) / t_0));
	} else {
		tmp = x / (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / y) + (y * -0.16666666666666666)
	tmp = 0
	if x <= 6.8e+153:
		tmp = x * (y / x)
	elif x <= 6.6e+180:
		tmp = (1.0 / t_0) + (-0.16666666666666666 * ((x * x) / t_0))
	else:
		tmp = x / (x / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / y) + Float64(y * -0.16666666666666666))
	tmp = 0.0
	if (x <= 6.8e+153)
		tmp = Float64(x * Float64(y / x));
	elseif (x <= 6.6e+180)
		tmp = Float64(Float64(1.0 / t_0) + Float64(-0.16666666666666666 * Float64(Float64(x * x) / t_0)));
	else
		tmp = Float64(x / Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / y) + (y * -0.16666666666666666);
	tmp = 0.0;
	if (x <= 6.8e+153)
		tmp = x * (y / x);
	elseif (x <= 6.6e+180)
		tmp = (1.0 / t_0) + (-0.16666666666666666 * ((x * x) / t_0));
	else
		tmp = x / (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / y), $MachinePrecision] + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 6.8e+153], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e+180], N[(N[(1.0 / t$95$0), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(x * x), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.7999999999999995e153

   1. Initial program 81.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 35.8%

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

   5. Taylor expanded in x around 0 17.2%

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

      1. associate-/l* 33.2%

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

      2. associate-/r/ 51.3%

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

   7. Applied egg-rr 51.3%

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

   if 6.7999999999999995e153 < x < 6.59999999999999978e180

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 50.4%

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

   5. Taylor expanded in x around 0 50.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{y} + -0.16666666666666666 \cdot y} + -0.16666666666666666 \cdot \frac{{x}^{2}}{\frac{1}{y} + -0.16666666666666666 \cdot y}} \]
   6. Step-by-step derivation

      1. *-commutative 50.7%

         \[\leadsto \frac{1}{\frac{1}{y} + \color{blue}{y \cdot -0.16666666666666666}} + -0.16666666666666666 \cdot \frac{{x}^{2}}{\frac{1}{y} + -0.16666666666666666 \cdot y} \]

      2. unpow2 50.7%

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

      3. *-commutative 50.7%

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

   7. Simplified 50.7%

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

   if 6.59999999999999978e180 < x

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. Taylor expanded in y around 0 57.0%

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

      1. associate-/l* 57.1%

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

   6. Simplified 57.1%

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

      1. associate-/l* 57.0%

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

      2. clear-num 56.9%

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

      3. associate-/r/ 57.0%

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

      4. *-commutative 57.0%

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

   8. Applied egg-rr 57.0%

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

   9. Taylor expanded in x around 0 7.8%

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

       1. associate-*l/ 7.8%

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

       2. *-un-lft-identity 7.8%

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

       3. *-commutative 7.8%

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

       4. associate-/l* 45.1%

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

   11. Applied egg-rr 45.1%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\frac{1}{y} + y \cdot -0.16666666666666666} + -0.16666666666666666 \cdot \frac{x \cdot x}{\frac{1}{y} + y \cdot -0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \]

Alternative 8: 50.1% 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 83.6%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. Taylor expanded in y around 0 38.1%

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

5. Taylor expanded in x around 0 16.8%

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

   1. associate-/l* 29.9%

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

   2. associate-/r/ 49.6%

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

7. Applied egg-rr 49.6%

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

8. Final simplification 49.6%

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

Alternative 9: 28.3% 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 83.6%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. Taylor expanded in y around 0 38.1%

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

   1. associate-/l* 54.5%

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

6. Simplified 54.5%

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

7. Taylor expanded in x around 0 29.9%

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

8. Final simplification 29.9%

   \[\leadsto y \]

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 2023242 
(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))