Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.5% → 99.9%

Time: 6.8s

Alternatives: 10

Speedup: 1.0×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.5% 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 87.3%

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

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

Alternative 2: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;\sinh y \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) (- INFINITY))
   (* (sinh y) (+ 1.0 (* (* x x) -0.16666666666666666)))
   (if (<= (sinh y) 2e-19) (* (/ (sin x) x) y) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -((double) INFINITY)) {
		tmp = sinh(y) * (1.0 + ((x * x) * -0.16666666666666666));
	} else if (sinh(y) <= 2e-19) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sinh(y) * (1.0 + ((x * x) * -0.16666666666666666));
	} else if (Math.sinh(y) <= 2e-19) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -math.inf:
		tmp = math.sinh(y) * (1.0 + ((x * x) * -0.16666666666666666))
	elif math.sinh(y) <= 2e-19:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= Float64(-Inf))
		tmp = Float64(sinh(y) * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	elseif (sinh(y) <= 2e-19)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= -Inf)
		tmp = sinh(y) * (1.0 + ((x * x) * -0.16666666666666666));
	elseif (sinh(y) <= 2e-19)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[(N[Sinh[y], $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 2e-19], 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) < -inf.0

   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 79.4%

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

      1. *-commutative 79.4%

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

      2. unpow2 79.4%

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

   6. Simplified 79.4%

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

   if -inf.0 < (sinh.f64 y) < 2e-19

   1. Initial program 72.8%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 99.9%

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

   if 2e-19 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. associate-*r/ 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in x around 0 85.0%

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

      1. add-log-exp 83.9%

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

      2. *-un-lft-identity 83.9%

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

      3. log-prod 83.9%

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

      4. metadata-eval 83.9%

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

      5. add-log-exp 85.0%

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

   6. Applied egg-rr 85.0%

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

      1. +-lft-identity 85.0%

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

      2. associate-*r/ 86.3%

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

      3. associate-*l/ 86.3%

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

      4. *-inverses 86.3%

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

      5. *-lft-identity 86.3%

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

   8. Simplified 86.3%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;\sinh y \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 3: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -3 \cdot 10^{-33}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-74}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -3e-33)
   (sinh y)
   (if (<= (sinh y) 1e-74) (/ x (/ x y)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -3e-33) {
		tmp = sinh(y);
	} else if (sinh(y) <= 1e-74) {
		tmp = x / (x / y);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sinh(y) <= (-3d-33)) then
        tmp = sinh(y)
    else if (sinh(y) <= 1d-74) then
        tmp = x / (x / y)
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -3e-33) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 1e-74) {
		tmp = x / (x / y);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -3e-33:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 1e-74:
		tmp = x / (x / y)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -3e-33)
		tmp = sinh(y);
	elseif (sinh(y) <= 1e-74)
		tmp = Float64(x / Float64(x / y));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= -3e-33)
		tmp = sinh(y);
	elseif (sinh(y) <= 1e-74)
		tmp = x / (x / y);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -3e-33], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 1e-74], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -3.0000000000000002e-33 or 9.99999999999999958e-75 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. associate-*r/ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 75.9%

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

      1. add-log-exp 73.8%

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

      2. *-un-lft-identity 73.8%

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

      3. log-prod 73.8%

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

      4. metadata-eval 73.8%

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

      5. add-log-exp 75.9%

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

   6. Applied egg-rr 75.9%

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

      1. +-lft-identity 75.9%

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

      2. associate-*r/ 76.6%

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

      3. associate-*l/ 76.6%

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

      4. *-inverses 76.6%

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

      5. *-lft-identity 76.6%

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

   8. Simplified 76.6%

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

   if -3.0000000000000002e-33 < (sinh.f64 y) < 9.99999999999999958e-75

   1. Initial program 69.9%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 69.9%

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

      1. *-commutative 69.9%

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

      2. associate-/l* 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in x around 0 78.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -3 \cdot 10^{-33}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-74}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. associate-*r/ 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 5: 86.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1040000000000:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1040000000000.0)
   (sinh y)
   (if (<= y 1.6e-11) (* (sin x) (/ y x)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1040000000000.0) {
		tmp = sinh(y);
	} else if (y <= 1.6e-11) {
		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 (y <= (-1040000000000.0d0)) then
        tmp = sinh(y)
    else if (y <= 1.6d-11) 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 (y <= -1040000000000.0) {
		tmp = Math.sinh(y);
	} else if (y <= 1.6e-11) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1040000000000.0:
		tmp = math.sinh(y)
	elif y <= 1.6e-11:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1040000000000.0)
		tmp = sinh(y);
	elseif (y <= 1.6e-11)
		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 (y <= -1040000000000.0)
		tmp = sinh(y);
	elseif (y <= 1.6e-11)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1040000000000.0], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 1.6e-11], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.04e12 or 1.59999999999999997e-11 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 80.8%

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

      1. add-log-exp 80.2%

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

      2. *-un-lft-identity 80.2%

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

      3. log-prod 80.2%

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

      4. metadata-eval 80.2%

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

      5. add-log-exp 80.8%

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

   6. Applied egg-rr 80.8%

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

      1. +-lft-identity 80.8%

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

      2. associate-*r/ 81.5%

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

      3. associate-*l/ 81.5%

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

      4. *-inverses 81.5%

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

      5. *-lft-identity 81.5%

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

   8. Simplified 81.5%

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

   if -1.04e12 < y < 1.59999999999999997e-11

   1. Initial program 73.1%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 99.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1040000000000:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 6: 86.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1040000000000:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1040000000000.0)
   (sinh y)
   (if (<= y 1.6e-11) (* (/ (sin x) x) y) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1040000000000.0) {
		tmp = sinh(y);
	} else if (y <= 1.6e-11) {
		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 <= (-1040000000000.0d0)) then
        tmp = sinh(y)
    else if (y <= 1.6d-11) 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 <= -1040000000000.0) {
		tmp = Math.sinh(y);
	} else if (y <= 1.6e-11) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1040000000000.0:
		tmp = math.sinh(y)
	elif y <= 1.6e-11:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1040000000000.0)
		tmp = sinh(y);
	elseif (y <= 1.6e-11)
		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 <= -1040000000000.0)
		tmp = sinh(y);
	elseif (y <= 1.6e-11)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1040000000000.0], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 1.6e-11], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.04e12 or 1.59999999999999997e-11 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 80.8%

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

      1. add-log-exp 80.2%

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

      2. *-un-lft-identity 80.2%

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

      3. log-prod 80.2%

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

      4. metadata-eval 80.2%

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

      5. add-log-exp 80.8%

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

   6. Applied egg-rr 80.8%

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

      1. +-lft-identity 80.8%

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

      2. associate-*r/ 81.5%

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

      3. associate-*l/ 81.5%

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

      4. *-inverses 81.5%

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

      5. *-lft-identity 81.5%

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

   8. Simplified 81.5%

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

   if -1.04e12 < y < 1.59999999999999997e-11

   1. Initial program 73.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1040000000000:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 7: 74.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. associate-*r/ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 76.1%

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

5. Final simplification 76.1%

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

Alternative 8: 51.2% accurate, 25.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. associate-*r/ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in y around 0 37.3%

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

   1. *-commutative 37.3%

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

   2. associate-/l* 60.3%

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

6. Simplified 60.3%

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

7. Taylor expanded in x around 0 48.4%

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

   1. frac-2neg 48.4%

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

   2. div-inv 50.9%

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

   3. distribute-neg-frac 50.9%

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

9. Applied egg-rr 50.9%

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

10. Final simplification 50.9%

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

Alternative 9: 50.5% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. associate-*r/ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 76.1%

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

5. Taylor expanded in y around 0 50.2%

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

6. Final simplification 50.2%

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

Alternative 10: 28.2% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 87.3%

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

   1. associate-*r/ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 76.1%

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

5. Taylor expanded in y around 0 29.0%

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

6. Final simplification 29.0%

   \[\leadsto y \]

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023224 
(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))