Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.7% → 99.9%

Time: 9.4s

Alternatives: 10

Speedup: 1.0×

• 49.6% 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: 88.7% 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 87.7%

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

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

Alternative 2: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2e+289) (* (sin x) (/ y x)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2e+289) {
		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+289) 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+289) {
		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+289:
		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+289)
		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+289)
		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+289], 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) < 2.0000000000000001e289

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*l/ 99.9%

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

      3. *-commutative 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 50.5%

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

      1. associate-/l* 67.6%

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

      2. associate-/r/ 78.3%

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

   6. Simplified 78.3%

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

   if 2.0000000000000001e289 < (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 72.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 3: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2e+289) (* (/ (sin x) x) y) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2e+289) {
		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+289) 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+289) {
		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+289:
		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+289)
		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+289)
		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+289], 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) < 2.0000000000000001e289

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*l/ 99.9%

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

      3. *-commutative 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 50.5%

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

      1. associate-/l* 67.6%

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

   6. Simplified 67.6%

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

      1. div-inv 67.6%

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

      2. *-commutative 67.6%

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

      3. clear-num 67.6%

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

   8. Applied egg-rr 67.6%

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

   if 2.0000000000000001e289 < (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 72.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 4: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2e+289) (/ y (/ x (sin x))) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2e+289) {
		tmp = y / (x / sin(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+289) then
        tmp = y / (x / sin(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+289) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 2e+289:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 2e+289)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 2e+289)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 2e+289], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 2.0000000000000001e289

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*l/ 99.9%

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

      3. *-commutative 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 50.5%

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

      1. associate-/l* 67.6%

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

   6. Simplified 67.6%

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

   if 2.0000000000000001e289 < (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 72.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 5: 57.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2e+289)
   (/ y (+ 1.0 (* x (* x 0.16666666666666666))))
   (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2e+289) {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	} 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+289) then
        tmp = y / (1.0d0 + (x * (x * 0.16666666666666666d0)))
    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+289) {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 2e+289:
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 2e+289)
		tmp = Float64(y / Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666))));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 2e+289)
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 2e+289], N[(y / N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 2.0000000000000001e289

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*l/ 99.9%

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

      3. *-commutative 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 50.5%

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

      1. associate-/l* 67.6%

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

      2. associate-/r/ 78.3%

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

   6. Simplified 78.3%

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

      1. associate-/r/ 67.6%

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

      2. div-inv 67.5%

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

      3. associate-/r* 78.2%

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

   8. Applied egg-rr 78.2%

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

   9. Taylor expanded in x around 0 65.3%

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

   10. Taylor expanded in y around 0 54.6%

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

       1. *-commutative 54.6%

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

       2. fma-udef 54.6%

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

       3. fma-udef 54.6%

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

       4. distribute-lft-in 54.6%

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

       5. *-commutative 54.6%

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

       6. +-commutative 54.6%

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

       7. rgt-mult-inverse 54.7%

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

       8. *-commutative 54.7%

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

   12. Simplified 54.7%

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

   if 2.0000000000000001e289 < (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 72.6%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

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 87.7%

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

   1. *-commutative 87.7%

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

   2. associate-*l/ 99.9%

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

   3. *-commutative 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: 49.0% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 38:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot 6}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 38.0)
   (/ y (+ 1.0 (* x (* x 0.16666666666666666))))
   (/ (* (/ y x) 6.0) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 38.0) {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = ((y / x) * 6.0) / 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 <= 38.0d0) then
        tmp = y / (1.0d0 + (x * (x * 0.16666666666666666d0)))
    else
        tmp = ((y / x) * 6.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 38.0) {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = ((y / x) * 6.0) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 38.0:
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)))
	else:
		tmp = ((y / x) * 6.0) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 38.0)
		tmp = Float64(y / Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666))));
	else
		tmp = Float64(Float64(Float64(y / x) * 6.0) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 38.0)
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	else
		tmp = ((y / x) * 6.0) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 38.0], N[(y / N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] * 6.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 38

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*l/ 99.9%

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

      3. *-commutative 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 50.8%

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

      1. associate-/l* 67.9%

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

      2. associate-/r/ 78.7%

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

   6. Simplified 78.7%

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

      1. associate-/r/ 67.9%

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

      2. div-inv 67.8%

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

      3. associate-/r* 78.6%

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

   8. Applied egg-rr 78.6%

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

   9. Taylor expanded in x around 0 65.6%

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

   10. Taylor expanded in y around 0 54.9%

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

       1. *-commutative 54.9%

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

       2. fma-udef 54.9%

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

       3. fma-udef 54.9%

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

       4. distribute-lft-in 54.9%

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

       5. *-commutative 54.9%

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

       6. +-commutative 54.9%

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

       7. rgt-mult-inverse 55.0%

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

       8. *-commutative 55.0%

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

   12. Simplified 55.0%

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

   if 38 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-commutative 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. Taylor expanded in y around 0 4.8%

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

      1. /-rgt-identity 4.8%

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

      2. associate-/r/ 4.8%

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

   6. Applied egg-rr 4.8%

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

   7. Taylor expanded in x around 0 3.9%

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

   8. Taylor expanded in x around inf 42.2%

      \[\leadsto \frac{\color{blue}{6 \cdot \frac{y}{x}}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 38:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot 6}{x}\\ \end{array} \]

Alternative 8: 54.2% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 38:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot 6}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 38.0) (/ x (/ x y)) (/ (* (/ y x) 6.0) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 38.0) {
		tmp = x / (x / y);
	} else {
		tmp = ((y / x) * 6.0) / 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 <= 38.0d0) then
        tmp = x / (x / y)
    else
        tmp = ((y / x) * 6.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 38.0) {
		tmp = x / (x / y);
	} else {
		tmp = ((y / x) * 6.0) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 38.0:
		tmp = x / (x / y)
	else:
		tmp = ((y / x) * 6.0) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 38.0)
		tmp = Float64(x / Float64(x / y));
	else
		tmp = Float64(Float64(Float64(y / x) * 6.0) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 38.0)
		tmp = x / (x / y);
	else
		tmp = ((y / x) * 6.0) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 38.0], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] * 6.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 38

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*l/ 99.9%

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

      3. *-commutative 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 50.8%

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

   5. Taylor expanded in x around 0 24.4%

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

      1. *-commutative 24.4%

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

   7. Simplified 24.4%

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

      1. associate-/l* 37.9%

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

      2. associate-/r/ 64.1%

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

   9. Applied egg-rr 64.1%

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

       1. *-commutative 64.1%

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

       2. clear-num 65.5%

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

       3. un-div-inv 64.6%

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

   11. Applied egg-rr 64.6%

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

   if 38 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-commutative 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. Taylor expanded in y around 0 4.8%

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

      1. /-rgt-identity 4.8%

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

      2. associate-/r/ 4.8%

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

   6. Applied egg-rr 4.8%

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

   7. Taylor expanded in x around 0 3.9%

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

   8. Taylor expanded in x around inf 42.2%

      \[\leadsto \frac{\color{blue}{6 \cdot \frac{y}{x}}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 38:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot 6}{x}\\ \end{array} \]

Alternative 9: 50.9% 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 87.7%

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

   1. *-commutative 87.7%

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

   2. associate-*l/ 99.9%

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

   3. *-commutative 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 37.5%

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

5. Taylor expanded in x around 0 20.3%

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

   1. *-commutative 20.3%

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

7. Simplified 20.3%

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

   1. associate-/l* 28.1%

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

   2. associate-/r/ 55.1%

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

9. Applied egg-rr 55.1%

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

10. Final simplification 55.1%

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

Alternative 10: 28.6% 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 87.7%

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

   1. *-commutative 87.7%

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

   2. associate-*l/ 99.9%

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

   3. *-commutative 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 37.5%

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

   1. associate-/l* 49.7%

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

   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 28.1%

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

8. Final simplification 28.1%

   \[\leadsto y \]

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 2023336 
(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))