Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.1% → 99.9%

Time: 9.0s

Alternatives: 13

Speedup: 1.0×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% 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 90.2%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.001:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -5e-5)
   (sinh y)
   (if (<= (sinh y) 0.001) (* (sin x) (/ y x)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -5e-5) {
		tmp = sinh(y);
	} else if (sinh(y) <= 0.001) {
		tmp = sin(x) * (y / x);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sinh(y) <= (-5d-5)) then
        tmp = sinh(y)
    else if (sinh(y) <= 0.001d0) then
        tmp = sin(x) * (y / x)
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -5e-5) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 0.001) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -5e-5:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 0.001:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -5e-5)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.001)
		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) <= -5e-5)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.001)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -5e-5], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.001], 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) < -5.00000000000000024e-5 or 1e-3 < (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 73.4%

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

   if -5.00000000000000024e-5 < (sinh.f64 y) < 1e-3

   1. Initial program 81.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 80.7%

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

      1. associate-/l* 99.5%

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

      2. associate-/r/ 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.001:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 3: 74.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.001:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -5e-5)
   (sinh y)
   (if (<= (sinh y) 0.001)
     (/ y (* x (+ (* x 0.16666666666666666) (/ 1.0 x))))
     (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -5e-5) {
		tmp = sinh(y);
	} else if (sinh(y) <= 0.001) {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sinh(y) <= (-5d-5)) then
        tmp = sinh(y)
    else if (sinh(y) <= 0.001d0) then
        tmp = y / (x * ((x * 0.16666666666666666d0) + (1.0d0 / x)))
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -5e-5) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 0.001) {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -5e-5:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 0.001:
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -5e-5)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.001)
		tmp = Float64(y / Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x))));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= -5e-5)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.001)
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -5e-5], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.001], N[(y / N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -5.00000000000000024e-5 or 1e-3 < (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 73.4%

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

   if -5.00000000000000024e-5 < (sinh.f64 y) < 1e-3

   1. Initial program 81.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 80.7%

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

      1. associate-/l* 99.5%

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

   6. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in x around 0 76.8%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.001:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 4: 59.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \frac{y}{x \cdot x}\\ t_1 := \sqrt{y \cdot y}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -245:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 245:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 6.0 (/ y (* x x)))) (t_1 (sqrt (* y y))))
   (if (<= y -1.7e+217)
     (* x (/ y x))
     (if (<= y -2.4e+155)
       t_1
       (if (<= y -245.0)
         t_0
         (if (<= y 245.0)
           (/ y (* x (+ (* x 0.16666666666666666) (/ 1.0 x))))
           (if (<= y 8e+150) t_0 t_1)))))))

C

double code(double x, double y) {
	double t_0 = 6.0 * (y / (x * x));
	double t_1 = sqrt((y * y));
	double tmp;
	if (y <= -1.7e+217) {
		tmp = x * (y / x);
	} else if (y <= -2.4e+155) {
		tmp = t_1;
	} else if (y <= -245.0) {
		tmp = t_0;
	} else if (y <= 245.0) {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	} else if (y <= 8e+150) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 6.0d0 * (y / (x * x))
    t_1 = sqrt((y * y))
    if (y <= (-1.7d+217)) then
        tmp = x * (y / x)
    else if (y <= (-2.4d+155)) then
        tmp = t_1
    else if (y <= (-245.0d0)) then
        tmp = t_0
    else if (y <= 245.0d0) then
        tmp = y / (x * ((x * 0.16666666666666666d0) + (1.0d0 / x)))
    else if (y <= 8d+150) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 6.0 * (y / (x * x));
	double t_1 = Math.sqrt((y * y));
	double tmp;
	if (y <= -1.7e+217) {
		tmp = x * (y / x);
	} else if (y <= -2.4e+155) {
		tmp = t_1;
	} else if (y <= -245.0) {
		tmp = t_0;
	} else if (y <= 245.0) {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	} else if (y <= 8e+150) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 6.0 * (y / (x * x))
	t_1 = math.sqrt((y * y))
	tmp = 0
	if y <= -1.7e+217:
		tmp = x * (y / x)
	elif y <= -2.4e+155:
		tmp = t_1
	elif y <= -245.0:
		tmp = t_0
	elif y <= 245.0:
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)))
	elif y <= 8e+150:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(6.0 * Float64(y / Float64(x * x)))
	t_1 = sqrt(Float64(y * y))
	tmp = 0.0
	if (y <= -1.7e+217)
		tmp = Float64(x * Float64(y / x));
	elseif (y <= -2.4e+155)
		tmp = t_1;
	elseif (y <= -245.0)
		tmp = t_0;
	elseif (y <= 245.0)
		tmp = Float64(y / Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x))));
	elseif (y <= 8e+150)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 6.0 * (y / (x * x));
	t_1 = sqrt((y * y));
	tmp = 0.0;
	if (y <= -1.7e+217)
		tmp = x * (y / x);
	elseif (y <= -2.4e+155)
		tmp = t_1;
	elseif (y <= -245.0)
		tmp = t_0;
	elseif (y <= 245.0)
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	elseif (y <= 8e+150)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(6.0 * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -1.7e+217], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.4e+155], t$95$1, If[LessEqual[y, -245.0], t$95$0, If[LessEqual[y, 245.0], N[(y / N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+150], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.6999999999999999e217

   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 y around 0 6.9%

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

   5. Taylor expanded in x around 0 34.1%

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

      1. associate-/l* 7.7%

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

      2. associate-/r/ 56.4%

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

   7. Applied egg-rr 56.4%

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

   if -1.6999999999999999e217 < y < -2.40000000000000021e155 or 7.99999999999999985e150 < 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 y around 0 4.5%

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

   5. Taylor expanded in x around 0 24.7%

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

      1. div-inv 24.7%

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

      2. associate-*l* 4.4%

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

      3. div-inv 4.4%

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

      4. *-inverses 4.4%

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

      5. *-commutative 4.4%

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

      6. *-un-lft-identity 4.4%

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

      7. add-sqr-sqrt 3.3%

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

      8. sqrt-unprod 58.6%

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

   7. Applied egg-rr 58.6%

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

   if -2.40000000000000021e155 < y < -245 or 245 < y < 7.99999999999999985e150

   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 y around 0 3.2%

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

      1. associate-/l* 3.2%

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

   6. Simplified 3.2%

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

      1. clear-num 3.2%

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

      2. associate-/r/ 3.2%

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

   8. Applied egg-rr 3.2%

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

   9. Taylor expanded in x around 0 2.6%

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

   10. Taylor expanded in x around inf 34.9%

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

       1. unpow2 34.9%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

   12. Simplified 34.9%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{x \cdot x}} \]

   if -245 < y < 245

   1. Initial program 81.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 80.2%

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

      1. associate-/l* 98.6%

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

   6. Simplified 98.6%

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

      1. clear-num 98.6%

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

      2. associate-/r/ 98.5%

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

   8. Applied egg-rr 98.5%

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

   9. Taylor expanded in x around 0 76.4%

      \[\leadsto \frac{y}{\color{blue}{\left(0.16666666666666666 \cdot x + \frac{1}{x}\right)} \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \mathbf{elif}\;y \leq -245:\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 245:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+150}:\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \]

Alternative 5: 86.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+217}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-5}:\\ \;\;\;\;\sinh y \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \leq 0.00195:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3e+217)
   (sinh y)
   (if (<= y -5.6e-5)
     (* (sinh y) (+ 1.0 (* -0.16666666666666666 (* x x))))
     (if (<= y 0.00195) (* (/ (sin x) x) y) (sinh y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3e+217) {
		tmp = sinh(y);
	} else if (y <= -5.6e-5) {
		tmp = sinh(y) * (1.0 + (-0.16666666666666666 * (x * x)));
	} else if (y <= 0.00195) {
		tmp = (sin(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 (y <= (-3d+217)) then
        tmp = sinh(y)
    else if (y <= (-5.6d-5)) then
        tmp = sinh(y) * (1.0d0 + ((-0.16666666666666666d0) * (x * x)))
    else if (y <= 0.00195d0) then
        tmp = (sin(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 (y <= -3e+217) {
		tmp = Math.sinh(y);
	} else if (y <= -5.6e-5) {
		tmp = Math.sinh(y) * (1.0 + (-0.16666666666666666 * (x * x)));
	} else if (y <= 0.00195) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3e+217:
		tmp = math.sinh(y)
	elif y <= -5.6e-5:
		tmp = math.sinh(y) * (1.0 + (-0.16666666666666666 * (x * x)))
	elif y <= 0.00195:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3e+217)
		tmp = sinh(y);
	elseif (y <= -5.6e-5)
		tmp = Float64(sinh(y) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x))));
	elseif (y <= 0.00195)
		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 (y <= -3e+217)
		tmp = sinh(y);
	elseif (y <= -5.6e-5)
		tmp = sinh(y) * (1.0 + (-0.16666666666666666 * (x * x)));
	elseif (y <= 0.00195)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3e+217], N[Sinh[y], $MachinePrecision], If[LessEqual[y, -5.6e-5], N[(N[Sinh[y], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00195], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.99999999999999976e217 or 0.0019499999999999999 < 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 81.1%

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

   if -2.99999999999999976e217 < y < -5.59999999999999992e-5

   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 82.0%

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

      1. unpow2 82.0%

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

   6. Simplified 82.0%

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

   if -5.59999999999999992e-5 < y < 0.0019499999999999999

   1. Initial program 81.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 80.7%

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

      1. associate-/l* 99.5%

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

   6. Simplified 99.5%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.5%

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

      3. clear-num 99.5%

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

   8. Applied egg-rr 99.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+217}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-5}:\\ \;\;\;\;\sinh y \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \leq 0.00195:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 6: 86.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 0.055:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.2e-6) (sinh y) (if (<= y 0.055) (* (/ (sin x) x) y) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.2e-6) {
		tmp = sinh(y);
	} else if (y <= 0.055) {
		tmp = (sin(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 (y <= (-4.2d-6)) then
        tmp = sinh(y)
    else if (y <= 0.055d0) then
        tmp = (sin(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 (y <= -4.2e-6) {
		tmp = Math.sinh(y);
	} else if (y <= 0.055) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.2e-6:
		tmp = math.sinh(y)
	elif y <= 0.055:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.2e-6)
		tmp = sinh(y);
	elseif (y <= 0.055)
		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 (y <= -4.2e-6)
		tmp = sinh(y);
	elseif (y <= 0.055)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.2e-6], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 0.055], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1999999999999996e-6 or 0.0550000000000000003 < 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 73.4%

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

   if -4.1999999999999996e-6 < y < 0.0550000000000000003

   1. Initial program 81.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 80.7%

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

      1. associate-/l* 99.5%

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

   6. Simplified 99.5%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.5%

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

      3. clear-num 99.5%

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

   8. Applied egg-rr 99.5%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 0.055:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 7: 55.3% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{+153}:\\ \;\;\;\;y + x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq -260 \lor \neg \left(y \leq 250\right):\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.5e+217)
   (* x (/ y x))
   (if (<= y -2e+153)
     (+ y (* x (* y (* x -0.16666666666666666))))
     (if (or (<= y -260.0) (not (<= y 250.0)))
       (* 6.0 (/ y (* x x)))
       (/ y (* x (+ (* x 0.16666666666666666) (/ 1.0 x))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.5e+217) {
		tmp = x * (y / x);
	} else if (y <= -2e+153) {
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	} else if ((y <= -260.0) || !(y <= 250.0)) {
		tmp = 6.0 * (y / (x * x));
	} else {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8.5d+217)) then
        tmp = x * (y / x)
    else if (y <= (-2d+153)) then
        tmp = y + (x * (y * (x * (-0.16666666666666666d0))))
    else if ((y <= (-260.0d0)) .or. (.not. (y <= 250.0d0))) then
        tmp = 6.0d0 * (y / (x * x))
    else
        tmp = y / (x * ((x * 0.16666666666666666d0) + (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.5e+217) {
		tmp = x * (y / x);
	} else if (y <= -2e+153) {
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	} else if ((y <= -260.0) || !(y <= 250.0)) {
		tmp = 6.0 * (y / (x * x));
	} else {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.5e+217:
		tmp = x * (y / x)
	elif y <= -2e+153:
		tmp = y + (x * (y * (x * -0.16666666666666666)))
	elif (y <= -260.0) or not (y <= 250.0):
		tmp = 6.0 * (y / (x * x))
	else:
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.5e+217)
		tmp = Float64(x * Float64(y / x));
	elseif (y <= -2e+153)
		tmp = Float64(y + Float64(x * Float64(y * Float64(x * -0.16666666666666666))));
	elseif ((y <= -260.0) || !(y <= 250.0))
		tmp = Float64(6.0 * Float64(y / Float64(x * x)));
	else
		tmp = Float64(y / Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.5e+217)
		tmp = x * (y / x);
	elseif (y <= -2e+153)
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	elseif ((y <= -260.0) || ~((y <= 250.0)))
		tmp = 6.0 * (y / (x * x));
	else
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.5e+217], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2e+153], N[(y + N[(x * N[(y * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -260.0], N[Not[LessEqual[y, 250.0]], $MachinePrecision]], N[(6.0 * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.50000000000000021e217

   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 y around 0 6.9%

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

   5. Taylor expanded in x around 0 34.1%

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

      1. associate-/l* 7.7%

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

      2. associate-/r/ 56.4%

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

   7. Applied egg-rr 56.4%

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

   if -8.50000000000000021e217 < y < -2e153

   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 y around 0 4.2%

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

      1. associate-/l* 4.2%

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

      2. associate-/r/ 17.7%

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

   6. Simplified 17.7%

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

   7. Taylor expanded in x around 0 44.9%

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

      1. expm1-log1p-u 44.9%

         \[\leadsto y + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)\right)} \]

      2. expm1-udef 44.9%

         \[\leadsto y + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)} - 1\right)} \]

      3. *-commutative 44.9%

         \[\leadsto y + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(y \cdot {x}^{2}\right) \cdot -0.16666666666666666}\right)} - 1\right) \]

      4. associate-*l* 44.9%

         \[\leadsto y + \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left({x}^{2} \cdot -0.16666666666666666\right)}\right)} - 1\right) \]

      5. unpow2 44.9%

         \[\leadsto y + \left(e^{\mathsf{log1p}\left(y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)\right)} - 1\right) \]

   9. Applied egg-rr 44.9%

      \[\leadsto y + \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 44.9%

          \[\leadsto y + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)\right)} \]

       2. expm1-log1p 44.9%

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

       3. *-commutative 44.9%

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

       4. associate-*l* 44.9%

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

       5. associate-*l* 44.9%

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

       6. *-commutative 44.9%

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

   11. Simplified 44.9%

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

   if -2e153 < y < -260 or 250 < 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 y around 0 3.6%

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

      1. associate-/l* 3.6%

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

   6. Simplified 3.6%

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

      1. clear-num 3.6%

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

      2. associate-/r/ 3.6%

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

   8. Applied egg-rr 3.6%

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

   9. Taylor expanded in x around 0 2.8%

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

   10. Taylor expanded in x around inf 32.1%

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

       1. unpow2 32.1%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

   12. Simplified 32.1%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{x \cdot x}} \]

   if -260 < y < 250

   1. Initial program 81.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 80.2%

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

      1. associate-/l* 98.6%

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

   6. Simplified 98.6%

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

      1. clear-num 98.6%

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

      2. associate-/r/ 98.5%

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

   8. Applied egg-rr 98.5%

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

   9. Taylor expanded in x around 0 76.4%

      \[\leadsto \frac{y}{\color{blue}{\left(0.16666666666666666 \cdot x + \frac{1}{x}\right)} \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{+153}:\\ \;\;\;\;y + x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq -260 \lor \neg \left(y \leq 250\right):\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \end{array} \]

Alternative 8: 55.3% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+155}:\\ \;\;\;\;y + x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq -135 \lor \neg \left(y \leq 200\right):\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot -0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.95e+217)
   (* x (/ y x))
   (if (<= y -1.1e+155)
     (+ y (* x (* y (* x -0.16666666666666666))))
     (if (or (<= y -135.0) (not (<= y 200.0)))
       (* 6.0 (/ y (* x x)))
       (/ y (+ 1.0 (* x (* x -0.16666666666666666))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.95e+217) {
		tmp = x * (y / x);
	} else if (y <= -1.1e+155) {
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	} else if ((y <= -135.0) || !(y <= 200.0)) {
		tmp = 6.0 * (y / (x * x));
	} else {
		tmp = y / (1.0 + (x * (x * -0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.95d+217)) then
        tmp = x * (y / x)
    else if (y <= (-1.1d+155)) then
        tmp = y + (x * (y * (x * (-0.16666666666666666d0))))
    else if ((y <= (-135.0d0)) .or. (.not. (y <= 200.0d0))) then
        tmp = 6.0d0 * (y / (x * x))
    else
        tmp = y / (1.0d0 + (x * (x * (-0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.95e+217) {
		tmp = x * (y / x);
	} else if (y <= -1.1e+155) {
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	} else if ((y <= -135.0) || !(y <= 200.0)) {
		tmp = 6.0 * (y / (x * x));
	} else {
		tmp = y / (1.0 + (x * (x * -0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.95e+217:
		tmp = x * (y / x)
	elif y <= -1.1e+155:
		tmp = y + (x * (y * (x * -0.16666666666666666)))
	elif (y <= -135.0) or not (y <= 200.0):
		tmp = 6.0 * (y / (x * x))
	else:
		tmp = y / (1.0 + (x * (x * -0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.95e+217)
		tmp = Float64(x * Float64(y / x));
	elseif (y <= -1.1e+155)
		tmp = Float64(y + Float64(x * Float64(y * Float64(x * -0.16666666666666666))));
	elseif ((y <= -135.0) || !(y <= 200.0))
		tmp = Float64(6.0 * Float64(y / Float64(x * x)));
	else
		tmp = Float64(y / Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.95e+217)
		tmp = x * (y / x);
	elseif (y <= -1.1e+155)
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	elseif ((y <= -135.0) || ~((y <= 200.0)))
		tmp = 6.0 * (y / (x * x));
	else
		tmp = y / (1.0 + (x * (x * -0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.95e+217], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.1e+155], N[(y + N[(x * N[(y * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -135.0], N[Not[LessEqual[y, 200.0]], $MachinePrecision]], N[(6.0 * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.94999999999999997e217

   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 y around 0 6.9%

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

   5. Taylor expanded in x around 0 34.1%

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

      1. associate-/l* 7.7%

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

      2. associate-/r/ 56.4%

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

   7. Applied egg-rr 56.4%

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

   if -1.94999999999999997e217 < y < -1.1000000000000001e155

   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 y around 0 4.2%

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

      1. associate-/l* 4.2%

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

      2. associate-/r/ 17.7%

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

   6. Simplified 17.7%

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

   7. Taylor expanded in x around 0 44.9%

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

      1. expm1-log1p-u 44.9%

         \[\leadsto y + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)\right)} \]

      2. expm1-udef 44.9%

         \[\leadsto y + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)} - 1\right)} \]

      3. *-commutative 44.9%

         \[\leadsto y + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(y \cdot {x}^{2}\right) \cdot -0.16666666666666666}\right)} - 1\right) \]

      4. associate-*l* 44.9%

         \[\leadsto y + \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left({x}^{2} \cdot -0.16666666666666666\right)}\right)} - 1\right) \]

      5. unpow2 44.9%

         \[\leadsto y + \left(e^{\mathsf{log1p}\left(y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)\right)} - 1\right) \]

   9. Applied egg-rr 44.9%

      \[\leadsto y + \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 44.9%

          \[\leadsto y + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)\right)} \]

       2. expm1-log1p 44.9%

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

       3. *-commutative 44.9%

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

       4. associate-*l* 44.9%

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

       5. associate-*l* 44.9%

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

       6. *-commutative 44.9%

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

   11. Simplified 44.9%

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

   if -1.1000000000000001e155 < y < -135 or 200 < 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 y around 0 3.6%

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

      1. associate-/l* 3.6%

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

   6. Simplified 3.6%

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

      1. clear-num 3.6%

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

      2. associate-/r/ 3.6%

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

   8. Applied egg-rr 3.6%

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

   9. Taylor expanded in x around 0 2.8%

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

   10. Taylor expanded in x around inf 32.1%

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

       1. unpow2 32.1%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

   12. Simplified 32.1%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{x \cdot x}} \]

   if -135 < y < 200

   1. Initial program 81.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 80.2%

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

      1. associate-/l* 98.6%

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

   6. Simplified 98.6%

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

      1. clear-num 98.6%

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

      2. associate-/r/ 98.5%

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

   8. Applied egg-rr 98.5%

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

   9. Taylor expanded in x around 0 76.4%

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

       1. *-commutative 76.4%

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

       2. distribute-lft-in 76.4%

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

       3. add-sqr-sqrt 38.8%

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

       4. sqrt-unprod 75.9%

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

       5. *-commutative 75.9%

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

       6. *-commutative 75.9%

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

       7. swap-sqr 75.9%

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

       8. metadata-eval 75.9%

          \[\leadsto \frac{y}{x \cdot \sqrt{\left(x \cdot x\right) \cdot \color{blue}{0.027777777777777776}} + x \cdot \frac{1}{x}} \]

       9. metadata-eval 75.9%

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

       10. swap-sqr 75.9%

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

       11. sqrt-unprod 37.1%

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

       12. add-sqr-sqrt 75.7%

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

       13. *-commutative 75.7%

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

       14. lft-mult-inverse 75.9%

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

   11. Applied egg-rr 75.9%

       \[\leadsto \frac{y}{\color{blue}{x \cdot \left(x \cdot -0.16666666666666666\right) + 1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+155}:\\ \;\;\;\;y + x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq -135 \lor \neg \left(y \leq 200\right):\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot -0.16666666666666666\right)}\\ \end{array} \]

Alternative 9: 54.7% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+218}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{+151}:\\ \;\;\;\;y + x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq -245 \lor \neg \left(y \leq 7 \cdot 10^{-11}\right):\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+218)
   (* x (/ y x))
   (if (<= y -1.45e+151)
     (+ y (* x (* y (* x -0.16666666666666666))))
     (if (or (<= y -245.0) (not (<= y 7e-11)))
       (* 6.0 (/ y (* x x)))
       (/ x (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+218) {
		tmp = x * (y / x);
	} else if (y <= -1.45e+151) {
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	} else if ((y <= -245.0) || !(y <= 7e-11)) {
		tmp = 6.0 * (y / (x * x));
	} else {
		tmp = x / (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.8d+218)) then
        tmp = x * (y / x)
    else if (y <= (-1.45d+151)) then
        tmp = y + (x * (y * (x * (-0.16666666666666666d0))))
    else if ((y <= (-245.0d0)) .or. (.not. (y <= 7d-11))) then
        tmp = 6.0d0 * (y / (x * x))
    else
        tmp = x / (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+218) {
		tmp = x * (y / x);
	} else if (y <= -1.45e+151) {
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	} else if ((y <= -245.0) || !(y <= 7e-11)) {
		tmp = 6.0 * (y / (x * x));
	} else {
		tmp = x / (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.8e+218:
		tmp = x * (y / x)
	elif y <= -1.45e+151:
		tmp = y + (x * (y * (x * -0.16666666666666666)))
	elif (y <= -245.0) or not (y <= 7e-11):
		tmp = 6.0 * (y / (x * x))
	else:
		tmp = x / (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+218)
		tmp = Float64(x * Float64(y / x));
	elseif (y <= -1.45e+151)
		tmp = Float64(y + Float64(x * Float64(y * Float64(x * -0.16666666666666666))));
	elseif ((y <= -245.0) || !(y <= 7e-11))
		tmp = Float64(6.0 * Float64(y / Float64(x * x)));
	else
		tmp = Float64(x / Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.8e+218)
		tmp = x * (y / x);
	elseif (y <= -1.45e+151)
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	elseif ((y <= -245.0) || ~((y <= 7e-11)))
		tmp = 6.0 * (y / (x * x));
	else
		tmp = x / (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+218], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.45e+151], N[(y + N[(x * N[(y * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -245.0], N[Not[LessEqual[y, 7e-11]], $MachinePrecision]], N[(6.0 * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.79999999999999961e218

   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 y around 0 6.9%

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

   5. Taylor expanded in x around 0 34.1%

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

      1. associate-/l* 7.7%

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

      2. associate-/r/ 56.4%

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

   7. Applied egg-rr 56.4%

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

   if -4.79999999999999961e218 < y < -1.45000000000000009e151

   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 y around 0 4.2%

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

      1. associate-/l* 4.2%

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

      2. associate-/r/ 17.7%

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

   6. Simplified 17.7%

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

   7. Taylor expanded in x around 0 44.9%

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

      1. expm1-log1p-u 44.9%

         \[\leadsto y + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)\right)} \]

      2. expm1-udef 44.9%

         \[\leadsto y + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)} - 1\right)} \]

      3. *-commutative 44.9%

         \[\leadsto y + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(y \cdot {x}^{2}\right) \cdot -0.16666666666666666}\right)} - 1\right) \]

      4. associate-*l* 44.9%

         \[\leadsto y + \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left({x}^{2} \cdot -0.16666666666666666\right)}\right)} - 1\right) \]

      5. unpow2 44.9%

         \[\leadsto y + \left(e^{\mathsf{log1p}\left(y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)\right)} - 1\right) \]

   9. Applied egg-rr 44.9%

      \[\leadsto y + \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 44.9%

          \[\leadsto y + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)\right)} \]

       2. expm1-log1p 44.9%

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

       3. *-commutative 44.9%

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

       4. associate-*l* 44.9%

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

       5. associate-*l* 44.9%

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

       6. *-commutative 44.9%

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

   11. Simplified 44.9%

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

   if -1.45000000000000009e151 < y < -245 or 7.00000000000000038e-11 < 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 y around 0 5.4%

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

      1. associate-/l* 5.4%

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

   6. Simplified 5.4%

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

      1. clear-num 5.4%

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

      2. associate-/r/ 5.4%

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

   8. Applied egg-rr 5.4%

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

   9. Taylor expanded in x around 0 2.9%

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

   10. Taylor expanded in x around inf 31.4%

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

       1. unpow2 31.4%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

   12. Simplified 31.4%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{x \cdot x}} \]

   if -245 < y < 7.00000000000000038e-11

   1. Initial program 81.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 80.3%

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

   5. Taylor expanded in x around 0 31.8%

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

      1. associate-/l* 50.4%

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

      2. associate-/r/ 76.0%

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

   7. Applied egg-rr 76.0%

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

      1. *-commutative 76.0%

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

      2. clear-num 76.6%

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

      3. un-div-inv 76.8%

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

   9. Applied egg-rr 76.8%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+218}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{+151}:\\ \;\;\;\;y + x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq -245 \lor \neg \left(y \leq 7 \cdot 10^{-11}\right):\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \]

Alternative 10: 55.6% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -252 \lor \neg \left(y \leq 7 \cdot 10^{-11}\right):\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -252.0) (not (<= y 7e-11)))
   (* 6.0 (/ y (* x x)))
   (/ x (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -252.0) || !(y <= 7e-11)) {
		tmp = 6.0 * (y / (x * x));
	} else {
		tmp = x / (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-252.0d0)) .or. (.not. (y <= 7d-11))) then
        tmp = 6.0d0 * (y / (x * x))
    else
        tmp = x / (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -252.0) || !(y <= 7e-11)) {
		tmp = 6.0 * (y / (x * x));
	} else {
		tmp = x / (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -252.0) or not (y <= 7e-11):
		tmp = 6.0 * (y / (x * x))
	else:
		tmp = x / (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -252.0) || !(y <= 7e-11))
		tmp = Float64(6.0 * Float64(y / Float64(x * x)));
	else
		tmp = Float64(x / Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -252.0) || ~((y <= 7e-11)))
		tmp = 6.0 * (y / (x * x));
	else
		tmp = x / (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -252.0], N[Not[LessEqual[y, 7e-11]], $MachinePrecision]], N[(6.0 * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -252 or 7.00000000000000038e-11 < 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 y around 0 5.6%

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

      1. associate-/l* 5.6%

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

   6. Simplified 5.6%

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

      1. clear-num 5.6%

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

      2. associate-/r/ 5.6%

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

   8. Applied egg-rr 5.6%

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

   9. Taylor expanded in x around 0 3.7%

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

   10. Taylor expanded in x around inf 35.1%

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

       1. unpow2 35.1%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

   12. Simplified 35.1%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{x \cdot x}} \]

   if -252 < y < 7.00000000000000038e-11

   1. Initial program 81.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 80.3%

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

   5. Taylor expanded in x around 0 31.8%

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

      1. associate-/l* 50.4%

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

      2. associate-/r/ 76.0%

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

   7. Applied egg-rr 76.0%

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

      1. *-commutative 76.0%

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

      2. clear-num 76.6%

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

      3. un-div-inv 76.8%

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

   9. Applied egg-rr 76.8%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -252 \lor \neg \left(y \leq 7 \cdot 10^{-11}\right):\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \]

Alternative 11: 47.7% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+148}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e+148) (* x (/ y x)) (/ (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.8e+148) {
		tmp = x * (y / x);
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.8d+148) then
        tmp = x * (y / x)
    else
        tmp = (x * y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e+148) {
		tmp = x * (y / x);
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.8e+148:
		tmp = x * (y / x)
	else:
		tmp = (x * y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.8e+148)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(Float64(x * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e+148)
		tmp = x * (y / x);
	else
		tmp = (x * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.8e+148], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.80000000000000003e148

   1. Initial program 89.3%

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

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

   5. Taylor expanded in x around 0 25.0%

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

      1. associate-/l* 30.4%

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

      2. associate-/r/ 54.5%

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

   7. Applied egg-rr 54.5%

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

   if 1.80000000000000003e148 < 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 y around 0 4.6%

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

   5. Taylor expanded in x around 0 38.7%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+148}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \]

Alternative 12: 49.7% 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 90.2%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 44.4%

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

5. Taylor expanded in x around 0 26.2%

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

   1. associate-/l* 28.3%

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

   2. associate-/r/ 51.4%

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

7. Applied egg-rr 51.4%

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

8. Final simplification 51.4%

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

Alternative 13: 27.3% 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 90.2%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 44.4%

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

   1. associate-/l* 54.1%

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

   2. associate-/r/ 63.8%

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

6. Simplified 63.8%

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

7. Taylor expanded in x around 0 28.3%

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

8. Final simplification 28.3%

   \[\leadsto y \]

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 2023214 
(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))