Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.8% → 99.9%

Time: 9.1s

Alternatives: 9

Speedup: 1.0×

• Could not converge on a ground truth

  1. reg0 = (bf "3.551819773755478614041195675136202008599e267")
  2. reg1 = (bf "2.357755110220035549154756609586017082245e235")

• 49.1% 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: 88.8% 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

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

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 90.3%

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

   1. associate-/l* 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. Add Preprocessing
5. Add Preprocessing

Alternative 2: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 2:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{x}{\sinh y}}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2.0) (* y (/ (sin x) x)) (/ 1.0 (/ (/ x (sinh y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2.0) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = 1.0 / ((x / sinh(y)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sinh(y) <= 2.0d0) then
        tmp = y * (sin(x) / x)
    else
        tmp = 1.0d0 / ((x / sinh(y)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= 2.0) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = 1.0 / ((x / Math.sinh(y)) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 2.0:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = 1.0 / ((x / math.sinh(y)) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 2.0)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(1.0 / Float64(Float64(x / sinh(y)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 2.0)
		tmp = y * (sin(x) / x);
	else
		tmp = 1.0 / ((x / sinh(y)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 2.0], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x / N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 2

   1. Initial program 87.1%

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

      1. associate-/l* 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. Add Preprocessing

   5. Taylor expanded in y around 0 53.7%

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

      1. associate-/l* 66.4%

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

   7. Simplified 66.4%

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

   if 2 < (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%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

      2. clear-num 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/r* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 71.9%

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

Alternative 3: 69.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2.0) (* y (/ (sin x) x)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2.0) {
		tmp = y * (sin(x) / 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) <= 2.0d0) then
        tmp = y * (sin(x) / 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) <= 2.0) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 2.0:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 2.0)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 2.0)
		tmp = y * (sin(x) / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 2.0], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 2

   1. Initial program 87.1%

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

      1. associate-/l* 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. Add Preprocessing

   5. Taylor expanded in y around 0 53.7%

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

      1. associate-/l* 66.4%

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

   7. Simplified 66.4%

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

   if 2 < (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%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

      2. clear-num 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/r* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 71.9%

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

      1. clear-num 71.9%

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

      2. *-un-lft-identity 71.9%

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

      3. div-inv 71.9%

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

      4. clear-num 71.9%

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

   9. Applied egg-rr 71.9%

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

       1. *-lft-identity 71.9%

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

       2. associate-*r/ 71.9%

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

       3. associate-*l/ 71.9%

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

       4. *-inverses 71.9%

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

       5. *-lft-identity 71.9%

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

   11. Simplified 71.9%

       \[\leadsto \color{blue}{\sinh y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 62.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2e-72) (/ x (/ x y)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2e-72) {
		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) <= 2d-72) 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) <= 2e-72) {
		tmp = x / (x / y);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 2e-72:
		tmp = x / (x / y)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 2e-72)
		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) <= 2e-72)
		tmp = x / (x / y);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 2e-72], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 1.9999999999999999e-72

   1. Initial program 86.4%

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

      1. associate-/l* 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. Add Preprocessing

   5. Taylor expanded in x around 0 72.7%

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

   6. Taylor expanded in y around 0 54.8%

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

      1. clear-num 56.6%

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

      2. un-div-inv 56.1%

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

   8. Applied egg-rr 56.1%

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

   if 1.9999999999999999e-72 < (sinh.f64 y)

   1. Initial program 98.7%

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

      1. associate-/l* 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. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 98.7%

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

      2. clear-num 98.7%

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

      3. *-commutative 98.7%

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

      4. associate-/r* 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 71.3%

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

      1. clear-num 71.4%

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

      2. *-un-lft-identity 71.4%

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

      3. div-inv 71.3%

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

      4. clear-num 71.3%

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

   9. Applied egg-rr 71.3%

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

       1. *-lft-identity 71.3%

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

       2. associate-*r/ 70.2%

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

       3. associate-*l/ 71.4%

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

       4. *-inverses 71.4%

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

       5. *-lft-identity 71.4%

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

   11. Simplified 71.4%

       \[\leadsto \color{blue}{\sinh y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.2% 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 90.3%

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

   1. associate-/l* 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. Add Preprocessing

5. Taylor expanded in x around 0 72.3%

   \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]
6. Add Preprocessing

Alternative 6: 51.1% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+88}:\\ \;\;\;\;y + y \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.5e-48)
   (/ x (/ x y))
   (if (<= y 1.85e+88)
     (+ y (* y (* -0.16666666666666666 (* x x))))
     (* x (* y (/ 1.0 x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-48) {
		tmp = x / (x / y);
	} else if (y <= 1.85e+88) {
		tmp = y + (y * (-0.16666666666666666 * (x * x)));
	} else {
		tmp = x * (y * (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.5d-48) then
        tmp = x / (x / y)
    else if (y <= 1.85d+88) then
        tmp = y + (y * ((-0.16666666666666666d0) * (x * x)))
    else
        tmp = x * (y * (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-48) {
		tmp = x / (x / y);
	} else if (y <= 1.85e+88) {
		tmp = y + (y * (-0.16666666666666666 * (x * x)));
	} else {
		tmp = x * (y * (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.5e-48:
		tmp = x / (x / y)
	elif y <= 1.85e+88:
		tmp = y + (y * (-0.16666666666666666 * (x * x)))
	else:
		tmp = x * (y * (1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.5e-48)
		tmp = Float64(x / Float64(x / y));
	elseif (y <= 1.85e+88)
		tmp = Float64(y + Float64(y * Float64(-0.16666666666666666 * Float64(x * x))));
	else
		tmp = Float64(x * Float64(y * Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.5e-48)
		tmp = x / (x / y);
	elseif (y <= 1.85e+88)
		tmp = y + (y * (-0.16666666666666666 * (x * x)));
	else
		tmp = x * (y * (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.5e-48], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e+88], N[(y + N[(y * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.4999999999999999e-48

   1. Initial program 86.4%

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

      1. associate-/l* 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. Add Preprocessing

   5. Taylor expanded in x around 0 73.2%

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

   6. Taylor expanded in y around 0 56.0%

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

      1. clear-num 57.7%

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

      2. un-div-inv 57.2%

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

   8. Applied egg-rr 57.2%

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

   if 2.4999999999999999e-48 < y < 1.84999999999999997e88

   1. Initial program 99.8%

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

      1. associate-/l* 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. Add Preprocessing

   5. Taylor expanded in y around 0 40.5%

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

   6. Taylor expanded in x around 0 36.3%

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

      1. associate-*r* 36.3%

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

   8. Simplified 36.3%

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

      1. unpow2 36.3%

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

   10. Applied egg-rr 36.3%

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

   if 1.84999999999999997e88 < y

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 32.8%

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

      1. clear-num 32.8%

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

      2. associate-/r/ 32.8%

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

   7. Applied egg-rr 32.8%

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

   8. Taylor expanded in x around 0 32.6%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+88}:\\ \;\;\;\;y + y \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.1% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.3%

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

   1. associate-/l* 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. Add Preprocessing

5. Taylor expanded in x around 0 72.3%

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

6. Taylor expanded in y around 0 48.8%

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

   1. clear-num 50.0%

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

   2. un-div-inv 48.9%

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

8. Applied egg-rr 48.9%

   \[\leadsto \color{blue}{\frac{x}{\frac{x}{y}}} \]
9. Add Preprocessing

Alternative 8: 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 90.3%

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

   1. associate-/l* 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. Add Preprocessing

5. Taylor expanded in x around 0 72.3%

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

6. Taylor expanded in y around 0 48.8%

   \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
7. Add Preprocessing

Alternative 9: 28.1% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 90.3%

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

   1. associate-/l* 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. Add Preprocessing

5. Taylor expanded in x around 0 72.3%

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

6. Taylor expanded in y around 0 48.8%

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

7. Taylor expanded in x around 0 27.4%

   \[\leadsto \color{blue}{y} \]
8. Add Preprocessing

Developer Target 1: 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 2024146 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

  (/ (* (sin x) (sinh y)) x))