Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.2% → 99.9%

Time: 7.6s

Alternatives: 9

Speedup: 1.0×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.1%

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

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

Alternative 2: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 10^{-11}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) (- INFINITY)) (not (<= (sinh y) 1e-11)))
   (sinh y)
   (* (sin x) (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -((double) INFINITY)) || !(sinh(y) <= 1e-11)) {
		tmp = sinh(y);
	} else {
		tmp = sin(x) * (y / x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -Double.POSITIVE_INFINITY) || !(Math.sinh(y) <= 1e-11)) {
		tmp = Math.sinh(y);
	} else {
		tmp = Math.sin(x) * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -math.inf) or not (math.sinh(y) <= 1e-11):
		tmp = math.sinh(y)
	else:
		tmp = math.sin(x) * (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= Float64(-Inf)) || !(sinh(y) <= 1e-11))
		tmp = sinh(y);
	else
		tmp = Float64(sin(x) * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -Inf) || ~((sinh(y) <= 1e-11)))
		tmp = sinh(y);
	else
		tmp = sin(x) * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 1e-11]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -inf.0 or 9.99999999999999939e-12 < (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 74.8%

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

   if -inf.0 < (sinh.f64 y) < 9.99999999999999939e-12

   1. Initial program 77.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 77.7%

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   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-/r/ 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 10^{-11}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \]

Alternative 3: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 10^{-11}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) (- INFINITY)) (not (<= (sinh y) 1e-11)))
   (sinh y)
   (* (/ (sin x) x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -((double) INFINITY)) || !(sinh(y) <= 1e-11)) {
		tmp = sinh(y);
	} 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) <= 1e-11)) {
		tmp = Math.sinh(y);
	} 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) <= 1e-11):
		tmp = math.sinh(y)
	else:
		tmp = (math.sin(x) / x) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= Float64(-Inf)) || !(sinh(y) <= 1e-11))
		tmp = sinh(y);
	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) <= 1e-11)))
		tmp = sinh(y);
	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], 1e-11]], $MachinePrecision]], N[Sinh[y], $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 9.99999999999999939e-12 < (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 74.8%

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

   if -inf.0 < (sinh.f64 y) < 9.99999999999999939e-12

   1. Initial program 77.7%

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

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 10^{-11}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \end{array} \]

Alternative 4: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 10^{-11}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) (- INFINITY)) (not (<= (sinh y) 1e-11)))
   (sinh y)
   (* y (/ 1.0 (+ 1.0 (* x (* x 0.16666666666666666)))))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -((double) INFINITY)) || !(sinh(y) <= 1e-11)) {
		tmp = sinh(y);
	} else {
		tmp = y * (1.0 / (1.0 + (x * (x * 0.16666666666666666))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -Double.POSITIVE_INFINITY) || !(Math.sinh(y) <= 1e-11)) {
		tmp = Math.sinh(y);
	} else {
		tmp = y * (1.0 / (1.0 + (x * (x * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -math.inf) or not (math.sinh(y) <= 1e-11):
		tmp = math.sinh(y)
	else:
		tmp = y * (1.0 / (1.0 + (x * (x * 0.16666666666666666))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= Float64(-Inf)) || !(sinh(y) <= 1e-11))
		tmp = sinh(y);
	else
		tmp = Float64(y * Float64(1.0 / Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -Inf) || ~((sinh(y) <= 1e-11)))
		tmp = sinh(y);
	else
		tmp = y * (1.0 / (1.0 + (x * (x * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 1e-11]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(y * N[(1.0 / N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -inf.0 or 9.99999999999999939e-12 < (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 74.8%

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

   if -inf.0 < (sinh.f64 y) < 9.99999999999999939e-12

   1. Initial program 77.7%

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

      1. *-commutative 77.7%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in y around 0 99.6%

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

   7. Taylor expanded in x around 0 75.9%

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

      1. clear-num 75.8%

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

      2. associate-/r/ 75.8%

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

      3. *-commutative 75.8%

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

      4. +-commutative 75.8%

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

      5. distribute-lft-in 75.8%

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

      6. *-commutative 75.8%

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

      7. lft-mult-inverse 76.0%

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

      8. *-commutative 76.0%

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

   9. Applied egg-rr 76.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 10^{-11}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \end{array} \]

Alternative 5: 56.5% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot 0.16666666666666666\right)\\ \mathbf{if}\;y \leq -12200 \lor \neg \left(y \leq 5200000\right):\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{1 + t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x 0.16666666666666666))))
   (if (or (<= y -12200.0) (not (<= y 5200000.0)))
     (/ y t_0)
     (* y (/ 1.0 (+ 1.0 t_0))))))

C

double code(double x, double y) {
	double t_0 = x * (x * 0.16666666666666666);
	double tmp;
	if ((y <= -12200.0) || !(y <= 5200000.0)) {
		tmp = y / t_0;
	} else {
		tmp = y * (1.0 / (1.0 + 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 = x * (x * 0.16666666666666666d0)
    if ((y <= (-12200.0d0)) .or. (.not. (y <= 5200000.0d0))) then
        tmp = y / t_0
    else
        tmp = y * (1.0d0 / (1.0d0 + t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x * 0.16666666666666666);
	double tmp;
	if ((y <= -12200.0) || !(y <= 5200000.0)) {
		tmp = y / t_0;
	} else {
		tmp = y * (1.0 / (1.0 + t_0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * 0.16666666666666666)
	tmp = 0
	if (y <= -12200.0) or not (y <= 5200000.0):
		tmp = y / t_0
	else:
		tmp = y * (1.0 / (1.0 + t_0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * 0.16666666666666666))
	tmp = 0.0
	if ((y <= -12200.0) || !(y <= 5200000.0))
		tmp = Float64(y / t_0);
	else
		tmp = Float64(y * Float64(1.0 / Float64(1.0 + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x * 0.16666666666666666);
	tmp = 0.0;
	if ((y <= -12200.0) || ~((y <= 5200000.0)))
		tmp = y / t_0;
	else
		tmp = y * (1.0 / (1.0 + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -12200.0], N[Not[LessEqual[y, 5200000.0]], $MachinePrecision]], N[(y / t$95$0), $MachinePrecision], N[(y * N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -12200 or 5.2e6 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 4.3%

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

   7. Taylor expanded in x around 0 3.6%

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

   8. Taylor expanded in x around inf 37.8%

      \[\leadsto \frac{y}{\color{blue}{\left(0.16666666666666666 \cdot x\right)} \cdot x} \]
   9. Step-by-step derivation

      1. *-commutative 37.8%

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

   10. Simplified 37.8%

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

   if -12200 < y < 5.2e6

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in y around 0 97.0%

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

   7. Taylor expanded in x around 0 74.0%

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

      1. clear-num 73.9%

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

      2. associate-/r/ 73.9%

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

      3. *-commutative 73.9%

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

      4. +-commutative 73.9%

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

      5. distribute-lft-in 73.9%

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

      6. *-commutative 73.9%

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

      7. lft-mult-inverse 74.1%

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

      8. *-commutative 74.1%

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

   9. Applied egg-rr 74.1%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12200 \lor \neg \left(y \leq 5200000\right):\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \end{array} \]

Alternative 6: 53.1% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-11}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4e-11)
   (/ y (* x (* x 0.16666666666666666)))
   (* x (/ 1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4e-11) {
		tmp = y / (x * (x * 0.16666666666666666));
	} else {
		tmp = x * (1.0 / (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4d-11)) then
        tmp = y / (x * (x * 0.16666666666666666d0))
    else
        tmp = x * (1.0d0 / (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4e-11) {
		tmp = y / (x * (x * 0.16666666666666666));
	} else {
		tmp = x * (1.0 / (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4e-11:
		tmp = y / (x * (x * 0.16666666666666666))
	else:
		tmp = x * (1.0 / (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4e-11)
		tmp = Float64(y / Float64(x * Float64(x * 0.16666666666666666)));
	else
		tmp = Float64(x * Float64(1.0 / Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4e-11)
		tmp = y / (x * (x * 0.16666666666666666));
	else
		tmp = x * (1.0 / (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4e-11], N[(y / N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.99999999999999976e-11

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 5.8%

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

   7. Taylor expanded in x around 0 3.8%

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

   8. Taylor expanded in x around inf 39.1%

      \[\leadsto \frac{y}{\color{blue}{\left(0.16666666666666666 \cdot x\right)} \cdot x} \]
   9. Step-by-step derivation

      1. *-commutative 39.1%

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

   10. Simplified 39.1%

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

   if -3.99999999999999976e-11 < y

   1. Initial program 84.8%

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

   2. Taylor expanded in y around 0 53.8%

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

   3. Taylor expanded in x around 0 23.5%

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

      1. frac-2neg 23.5%

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

      2. div-inv 23.5%

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

      3. distribute-rgt-neg-in 23.5%

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

   5. Applied egg-rr 23.5%

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

      1. associate-*l* 59.4%

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

   7. Simplified 59.4%

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

      1. un-div-inv 59.5%

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

      2. frac-2neg 59.5%

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

      3. clear-num 60.3%

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

   9. Applied egg-rr 60.3%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-11}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \end{array} \]

Alternative 7: 50.4% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{1}{\frac{x}{y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ 1.0 (/ x y))))

C

double code(double x, double y) {
	return x * (1.0 / (x / y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 / (x / y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 / (x / y));
}

Python

def code(x, y):
	return x * (1.0 / (x / y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 / Float64(x / y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 / (x / y));
end

Wolfram

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

Derivation

1. Initial program 89.1%

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

2. Taylor expanded in y around 0 40.3%

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

3. Taylor expanded in x around 0 21.8%

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

   1. frac-2neg 21.8%

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

   2. div-inv 21.7%

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

   3. distribute-rgt-neg-in 21.7%

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

5. Applied egg-rr 21.7%

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

   1. associate-*l* 50.8%

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

7. Simplified 50.8%

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

   1. un-div-inv 50.8%

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

   2. frac-2neg 50.8%

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

   3. clear-num 51.4%

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

9. Applied egg-rr 51.4%

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

10. Final simplification 51.4%

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

Alternative 8: 49.8% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.1%

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

2. Taylor expanded in y around 0 40.3%

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

3. Taylor expanded in x around 0 21.8%

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

   1. associate-/l* 49.9%

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

   2. div-inv 51.4%

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

   3. clear-num 50.8%

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

5. Applied egg-rr 50.8%

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

6. Final simplification 50.8%

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

Alternative 9: 27.9% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 89.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 y around 0 51.2%

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

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 2023318 
(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))