Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.1% → 99.9%

Time: 13.4s

Alternatives: 13

Speedup: 1.0×

• 49.9% 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: 88.1% 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 89.5%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. Final simplification 100.0%

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

Alternative 2: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\frac{\frac{\sin x}{x}}{y \cdot -0.16666666666666666 + \frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) (- INFINITY))
   (sinh y)
   (if (<= (sinh y) 0.005)
     (/ (/ (sin x) x) (+ (* y -0.16666666666666666) (/ 1.0 y)))
     (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -((double) INFINITY)) {
		tmp = sinh(y);
	} else if (sinh(y) <= 0.005) {
		tmp = (sin(x) / x) / ((y * -0.16666666666666666) + (1.0 / y));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 0.005) {
		tmp = (Math.sin(x) / x) / ((y * -0.16666666666666666) + (1.0 / y));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -math.inf:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 0.005:
		tmp = (math.sin(x) / x) / ((y * -0.16666666666666666) + (1.0 / y))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= Float64(-Inf))
		tmp = sinh(y);
	elseif (sinh(y) <= 0.005)
		tmp = Float64(Float64(sin(x) / x) / Float64(Float64(y * -0.16666666666666666) + Float64(1.0 / y)));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= -Inf)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.005)
		tmp = (sin(x) / x) / ((y * -0.16666666666666666) + (1.0 / y));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.005], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] / N[(N[(y * -0.16666666666666666), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -inf.0 or 0.0050000000000000001 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l* 100.0%

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

      3. div-inv 100.0%

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

      4. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 81.4%

      \[\leadsto \frac{\color{blue}{1}}{\frac{1}{\sinh y}} \]
   7. Step-by-step derivation

      1. remove-double-div 81.4%

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

      2. expm1-log1p-u 40.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y\right)\right)} \]

      3. expm1-udef 40.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y\right)} - 1} \]

   8. Applied egg-rr 40.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 40.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y\right)\right)} \]

      2. expm1-log1p 81.4%

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

   10. Simplified 81.4%

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

   if -inf.0 < (sinh.f64 y) < 0.0050000000000000001

   1. Initial program 76.9%

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

      1. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 76.9%

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

      2. associate-/l* 98.8%

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

      3. div-inv 98.6%

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

      4. associate-/r* 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 99.6%

      \[\leadsto \frac{\frac{\sin x}{x}}{\color{blue}{-0.16666666666666666 \cdot y + \frac{1}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\frac{\frac{\sin x}{x}}{y \cdot -0.16666666666666666 + \frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 3: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -2e-6)
   (sinh y)
   (if (<= (sinh y) 0.005) (* (sin x) (/ y x)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -2e-6) {
		tmp = sinh(y);
	} else if (sinh(y) <= 0.005) {
		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) <= (-2d-6)) then
        tmp = sinh(y)
    else if (sinh(y) <= 0.005d0) 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) <= -2e-6) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 0.005) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -2e-6:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 0.005:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -2e-6)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.005)
		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) <= -2e-6)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.005)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -2e-6], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.005], 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) < -1.99999999999999991e-6 or 0.0050000000000000001 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l* 100.0%

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

      3. div-inv 100.0%

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

      4. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 81.5%

      \[\leadsto \frac{\color{blue}{1}}{\frac{1}{\sinh y}} \]
   7. Step-by-step derivation

      1. remove-double-div 81.5%

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

      2. expm1-log1p-u 41.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y\right)\right)} \]

      3. expm1-udef 40.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y\right)} - 1} \]

   8. Applied egg-rr 40.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 41.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y\right)\right)} \]

      2. expm1-log1p 81.5%

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

   10. Simplified 81.5%

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

   if -1.99999999999999991e-6 < (sinh.f64 y) < 0.0050000000000000001

   1. Initial program 76.7%

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

      1. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 76.2%

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

      1. associate-/l* 99.4%

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

      2. associate-/r/ 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 4: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -2e-6)
   (sinh y)
   (if (<= (sinh y) 0.005) (* (/ (sin x) x) y) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -2e-6) {
		tmp = sinh(y);
	} else if (sinh(y) <= 0.005) {
		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 (sinh(y) <= (-2d-6)) then
        tmp = sinh(y)
    else if (sinh(y) <= 0.005d0) 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 (Math.sinh(y) <= -2e-6) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 0.005) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -2e-6:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 0.005:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -2e-6)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.005)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= -2e-6)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.005)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -2e-6], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.005], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -1.99999999999999991e-6 or 0.0050000000000000001 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l* 100.0%

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

      3. div-inv 100.0%

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

      4. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 81.5%

      \[\leadsto \frac{\color{blue}{1}}{\frac{1}{\sinh y}} \]
   7. Step-by-step derivation

      1. remove-double-div 81.5%

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

      2. expm1-log1p-u 41.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y\right)\right)} \]

      3. expm1-udef 40.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y\right)} - 1} \]

   8. Applied egg-rr 40.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 41.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y\right)\right)} \]

      2. expm1-log1p 81.5%

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

   10. Simplified 81.5%

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

   if -1.99999999999999991e-6 < (sinh.f64 y) < 0.0050000000000000001

   1. Initial program 76.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.5%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 5: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{1}{\sinh y}}\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -2e-6)
   (/ 1.0 (/ 1.0 (sinh y)))
   (if (<= (sinh y) 0.005) (* (/ (sin x) x) y) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -2e-6) {
		tmp = 1.0 / (1.0 / sinh(y));
	} else if (sinh(y) <= 0.005) {
		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 (sinh(y) <= (-2d-6)) then
        tmp = 1.0d0 / (1.0d0 / sinh(y))
    else if (sinh(y) <= 0.005d0) 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 (Math.sinh(y) <= -2e-6) {
		tmp = 1.0 / (1.0 / Math.sinh(y));
	} else if (Math.sinh(y) <= 0.005) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -2e-6:
		tmp = 1.0 / (1.0 / math.sinh(y))
	elif math.sinh(y) <= 0.005:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -2e-6)
		tmp = Float64(1.0 / Float64(1.0 / sinh(y)));
	elseif (sinh(y) <= 0.005)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= -2e-6)
		tmp = 1.0 / (1.0 / sinh(y));
	elseif (sinh(y) <= 0.005)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -2e-6], N[(1.0 / N[(1.0 / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.005], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sinh.f64 y) < -1.99999999999999991e-6

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l* 100.0%

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

      3. div-inv 100.0%

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

      4. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 86.6%

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

   if -1.99999999999999991e-6 < (sinh.f64 y) < 0.0050000000000000001

   1. Initial program 76.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.5%

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

   if 0.0050000000000000001 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l* 100.0%

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

      3. div-inv 100.0%

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

      4. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 77.0%

      \[\leadsto \frac{\color{blue}{1}}{\frac{1}{\sinh y}} \]
   7. Step-by-step derivation

      1. remove-double-div 77.0%

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

      2. expm1-log1p-u 76.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y\right)\right)} \]

      3. expm1-udef 76.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y\right)} - 1} \]

   8. Applied egg-rr 76.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 76.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y\right)\right)} \]

      2. expm1-log1p 77.0%

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

   10. Simplified 77.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{1}{\sinh y}}\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.5%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 7: 65.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+129}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 2.1e+129) (sinh y) (sqrt (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.1e+129) {
		tmp = sinh(y);
	} else {
		tmp = sqrt((y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.1d+129) then
        tmp = sinh(y)
    else
        tmp = sqrt((y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.1e+129) {
		tmp = Math.sinh(y);
	} else {
		tmp = Math.sqrt((y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.1e+129:
		tmp = math.sinh(y)
	else:
		tmp = math.sqrt((y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.1e+129)
		tmp = sinh(y);
	else
		tmp = sqrt(Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.1e+129)
		tmp = sinh(y);
	else
		tmp = sqrt((y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.1e+129], N[Sinh[y], $MachinePrecision], N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.09999999999999997e129

   1. Initial program 87.7%

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

      1. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*r/ 87.7%

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

      2. associate-/l* 99.5%

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

      3. div-inv 99.4%

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

      4. associate-/r* 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 76.2%

      \[\leadsto \frac{\color{blue}{1}}{\frac{1}{\sinh y}} \]
   7. Step-by-step derivation

      1. remove-double-div 76.3%

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

      2. expm1-log1p-u 51.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y\right)\right)} \]

      3. expm1-udef 31.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y\right)} - 1} \]

   8. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 51.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y\right)\right)} \]

      2. expm1-log1p 76.3%

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

   10. Simplified 76.3%

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

   if 2.09999999999999997e129 < x

   1. Initial program 99.9%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 99.9%

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

      2. associate-/l* 99.0%

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

      3. div-inv 99.1%

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

      4. associate-/r* 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 29.3%

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

   7. Taylor expanded in y around 0 4.4%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{y}}} \]
   8. Step-by-step derivation

      1. remove-double-div 4.4%

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

      2. add-sqr-sqrt 2.7%

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

      3. sqrt-unprod 45.2%

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

   9. Applied egg-rr 45.2%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+129}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \]

Alternative 8: 63.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return sinh(y)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Sinh[y], $MachinePrecision]

Derivation

1. Initial program 89.5%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

   1. associate-*r/ 89.5%

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

   2. associate-/l* 99.4%

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

   3. div-inv 99.4%

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

   4. associate-/r* 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in x around 0 69.2%

   \[\leadsto \frac{\color{blue}{1}}{\frac{1}{\sinh y}} \]
7. Step-by-step derivation

   1. remove-double-div 69.3%

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

   2. expm1-log1p-u 47.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y\right)\right)} \]

   3. expm1-udef 35.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y\right)} - 1} \]

8. Applied egg-rr 35.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 47.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y\right)\right)} \]

   2. expm1-log1p 69.3%

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

10. Simplified 69.3%

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

11. Final simplification 69.3%

    \[\leadsto \sinh y \]

Alternative 9: 35.8% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.52 \cdot 10^{+181}:\\ \;\;\;\;y + -0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + -0.16666666666666666 \cdot \left(x \cdot x\right)}{y \cdot -0.16666666666666666 + \frac{1}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.52e+181)
   (+ y (* -0.16666666666666666 (* y (* x x))))
   (/
    (+ 1.0 (* -0.16666666666666666 (* x x)))
    (+ (* y -0.16666666666666666) (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.52e+181) {
		tmp = y + (-0.16666666666666666 * (y * (x * x)));
	} else {
		tmp = (1.0 + (-0.16666666666666666 * (x * x))) / ((y * -0.16666666666666666) + (1.0 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.52d+181) then
        tmp = y + ((-0.16666666666666666d0) * (y * (x * x)))
    else
        tmp = (1.0d0 + ((-0.16666666666666666d0) * (x * x))) / ((y * (-0.16666666666666666d0)) + (1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.52e+181) {
		tmp = y + (-0.16666666666666666 * (y * (x * x)));
	} else {
		tmp = (1.0 + (-0.16666666666666666 * (x * x))) / ((y * -0.16666666666666666) + (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.52e+181:
		tmp = y + (-0.16666666666666666 * (y * (x * x)))
	else:
		tmp = (1.0 + (-0.16666666666666666 * (x * x))) / ((y * -0.16666666666666666) + (1.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.52e+181)
		tmp = Float64(y + Float64(-0.16666666666666666 * Float64(y * Float64(x * x))));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x))) / Float64(Float64(y * -0.16666666666666666) + Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.52e+181)
		tmp = y + (-0.16666666666666666 * (y * (x * x)));
	else
		tmp = (1.0 + (-0.16666666666666666 * (x * x))) / ((y * -0.16666666666666666) + (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.52e+181], N[(y + N[(-0.16666666666666666 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y * -0.16666666666666666), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.52e181

   1. Initial program 88.2%

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

      1. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 36.0%

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

      1. associate-/l* 47.7%

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

      2. associate-/r/ 65.7%

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

   6. Simplified 65.7%

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

   7. Taylor expanded in x around 0 34.8%

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

      1. *-commutative 34.8%

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

      2. unpow2 34.8%

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

   9. Simplified 34.8%

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

   if 1.52e181 < x

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. associate-/l* 99.9%

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

      3. div-inv 99.9%

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

      4. associate-/r* 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around 0 45.6%

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

   7. Taylor expanded in x around 0 31.7%

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

      1. *-commutative 24.8%

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

      2. unpow2 24.8%

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

   9. Simplified 31.7%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.52 \cdot 10^{+181}:\\ \;\;\;\;y + -0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + -0.16666666666666666 \cdot \left(x \cdot x\right)}{y \cdot -0.16666666666666666 + \frac{1}{y}}\\ \end{array} \]

Alternative 10: 31.9% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+78}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.4e+78) y (* -0.16666666666666666 (* x (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.4e+78) {
		tmp = y;
	} else {
		tmp = -0.16666666666666666 * (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 (x <= 2.4d+78) then
        tmp = y
    else
        tmp = (-0.16666666666666666d0) * (x * (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.4e+78) {
		tmp = y;
	} else {
		tmp = -0.16666666666666666 * (x * (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.4e+78:
		tmp = y
	else:
		tmp = -0.16666666666666666 * (x * (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.4e+78)
		tmp = y;
	else
		tmp = Float64(-0.16666666666666666 * Float64(x * Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.4e+78)
		tmp = y;
	else
		tmp = -0.16666666666666666 * (x * (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.4e+78], y, N[(-0.16666666666666666 * N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.3999999999999999e78

   1. Initial program 87.1%

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

      1. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 33.4%

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

      1. associate-/l* 46.2%

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

      2. associate-/r/ 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in x around 0 32.6%

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

   if 2.3999999999999999e78 < x

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 53.4%

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

   5. Taylor expanded in x around 0 20.1%

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

      1. *-commutative 20.1%

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

      2. unpow2 20.1%

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

   7. Simplified 20.1%

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

   8. Taylor expanded in x around inf 20.1%

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

      1. unpow2 20.1%

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

      2. *-commutative 20.1%

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

      3. *-commutative 20.1%

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

      4. associate-*r* 20.1%

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

      5. associate-*r* 20.1%

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

      6. *-commutative 20.1%

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

   10. Simplified 20.1%

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

   11. Taylor expanded in y around 0 20.1%

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

       1. unpow2 20.1%

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

       2. associate-*l* 20.1%

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

       3. *-commutative 20.1%

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

   13. Simplified 20.1%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+78}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot y\right)\right)\\ \end{array} \]

Alternative 11: 36.0% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* y (+ 1.0 (* -0.16666666666666666 (* x x)))))

C

double code(double x, double y) {
	return y * (1.0 + (-0.16666666666666666 * (x * x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * (1.0d0 + ((-0.16666666666666666d0) * (x * x)))
end function

Java

public static double code(double x, double y) {
	return y * (1.0 + (-0.16666666666666666 * (x * x)));
}

Python

def code(x, y):
	return y * (1.0 + (-0.16666666666666666 * (x * x)))

Julia

function code(x, y)
	return Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x))))
end

MATLAB

function tmp = code(x, y)
	tmp = y * (1.0 + (-0.16666666666666666 * (x * x)));
end

Wolfram

code[x_, y_] := N[(y * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.5%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 47.6%

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

5. Taylor expanded in x around 0 33.6%

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

   1. *-commutative 33.6%

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

   2. unpow2 33.6%

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

7. Simplified 33.6%

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

8. Final simplification 33.6%

   \[\leadsto y \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \]

Alternative 12: 36.1% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ y + -0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ y (* -0.16666666666666666 (* y (* x x)))))

C

double code(double x, double y) {
	return y + (-0.16666666666666666 * (y * (x * x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y + ((-0.16666666666666666d0) * (y * (x * x)))
end function

Java

public static double code(double x, double y) {
	return y + (-0.16666666666666666 * (y * (x * x)));
}

Python

def code(x, y):
	return y + (-0.16666666666666666 * (y * (x * x)))

Julia

function code(x, y)
	return Float64(y + Float64(-0.16666666666666666 * Float64(y * Float64(x * x))))
end

MATLAB

function tmp = code(x, y)
	tmp = y + (-0.16666666666666666 * (y * (x * x)));
end

Wolfram

code[x_, y_] := N[(y + N[(-0.16666666666666666 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.5%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around 0 37.2%

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

   1. associate-/l* 47.6%

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

   2. associate-/r/ 63.5%

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

6. Simplified 63.5%

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

7. Taylor expanded in x around 0 33.6%

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

   1. *-commutative 33.6%

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

   2. unpow2 33.6%

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

9. Simplified 33.6%

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

10. Final simplification 33.6%

    \[\leadsto y + -0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right) \]

Alternative 13: 28.0% 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 89.5%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around 0 37.2%

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

   1. associate-/l* 47.6%

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

   2. associate-/r/ 63.5%

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

6. Simplified 63.5%

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

7. Taylor expanded in x around 0 27.3%

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

8. Final simplification 27.3%

   \[\leadsto y \]

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 2023287 
(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))