Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.6% → 99.9%

Time: 14.4s

Alternatives: 18

Speedup: 1.6×

• 49.4% 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: 88.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.8%

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

   1. *-commutative N/A

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

   2. associate-/r/ N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\sinh y, \color{blue}{\left(\frac{x}{\sin x}\right)}\right) \]

   4. sinh-lowering-sinh.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \left(\frac{\color{blue}{x}}{\sin x}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \mathsf{/.f64}\left(x, \color{blue}{\sin x}\right)\right) \]

   6. sin-lowering-sin.f64 99.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right) \]

6. Applied egg-rr 99.9%

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

Alternative 2: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 10^{-7}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 1e-7) (* (sin x) (/ y x)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 1e-7) {
		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) <= 1d-7) 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) <= 1e-7) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 1e-7:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 1e-7)
		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) <= 1e-7)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 1e-7], 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) < 9.9999999999999995e-8

   1. Initial program 83.5%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \color{blue}{\left(\frac{y}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 77.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   7. Simplified 77.5%

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

   if 9.9999999999999995e-8 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 79.4%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

         \[\leadsto \sinh y \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 79.4%

         \[\leadsto \mathsf{sinh.f64}\left(y\right) \]

   9. Applied egg-rr 79.4%

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

Alternative 3: 57.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 10^{-7}:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 1e-7)
   (/ y (+ 1.0 (* x (* x 0.16666666666666666))))
   (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 1e-7) {
		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) <= 1d-7) 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) <= 1e-7) {
		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) <= 1e-7:
		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) <= 1e-7)
		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) <= 1e-7)
		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], 1e-7], 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) < 9.9999999999999995e-8

   1. Initial program 83.5%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

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

      1. *-commutative N/A

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

      2. associate-/r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sinh y, \color{blue}{\left(\frac{x}{\sin x}\right)}\right) \]

      4. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \left(\frac{\color{blue}{x}}{\sin x}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \mathsf{/.f64}\left(x, \color{blue}{\sin x}\right)\right) \]

      6. sin-lowering-sin.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{y}, \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 69.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(1 + \frac{1}{6} \cdot {x}^{2}\right)}\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       6. *-lowering-*.f64 53.5%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   12. Simplified 53.5%

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

   if 9.9999999999999995e-8 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 79.4%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

         \[\leadsto \sinh y \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 79.4%

         \[\leadsto \mathsf{sinh.f64}\left(y\right) \]

   9. Applied egg-rr 79.4%

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

Alternative 4: 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 87.6%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.8%

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

Alternative 5: 89.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9.2e-7)
   (* y (* (+ 1.0 (* 0.16666666666666666 (* y y))) (/ (sin x) x)))
   (if (<= y 1.15e+62)
     (sinh y)
     (/
      (*
       y
       (*
        (sin x)
        (+
         1.0
         (*
          y
          (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))))
      x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-7) {
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (sin(x) / x));
	} else if (y <= 1.15e+62) {
		tmp = sinh(y);
	} else {
		tmp = (y * (sin(x) * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / 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 <= 9.2d-7) then
        tmp = y * ((1.0d0 + (0.16666666666666666d0 * (y * y))) * (sin(x) / x))
    else if (y <= 1.15d+62) then
        tmp = sinh(y)
    else
        tmp = (y * (sin(x) * (1.0d0 + (y * (y * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0))))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-7) {
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (Math.sin(x) / x));
	} else if (y <= 1.15e+62) {
		tmp = Math.sinh(y);
	} else {
		tmp = (y * (Math.sin(x) * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.2e-7:
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (math.sin(x) / x))
	elif y <= 1.15e+62:
		tmp = math.sinh(y)
	else:
		tmp = (y * (math.sin(x) * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.2e-7)
		tmp = Float64(y * Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(sin(x) / x)));
	elseif (y <= 1.15e+62)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(y * Float64(sin(x) * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.2e-7)
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (sin(x) / x));
	elseif (y <= 1.15e+62)
		tmp = sinh(y);
	else
		tmp = (y * (sin(x) * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.2e-7], N[(y * N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+62], N[Sinh[y], $MachinePrecision], N[(N[(y * N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.1999999999999998e-7

   1. Initial program 83.5%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 87.0%

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

   if 9.1999999999999998e-7 < y < 1.14999999999999992e62

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 87.5%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

         \[\leadsto \sinh y \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 87.5%

         \[\leadsto \mathsf{sinh.f64}\left(y\right) \]

   9. Applied egg-rr 87.5%

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

   if 1.14999999999999992e62 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right), x\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x\right)\right)\right)\right), x\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \sin x\right)\right)\right), x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left(\frac{1}{6} \cdot {y}^{2} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      11. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 88.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 0.0064:\\ \;\;\;\;y \cdot \left(t\_0 \cdot \frac{\sin x}{x}\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sinh y \cdot x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot t\_0\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (<= y 0.0064)
     (* y (* t_0 (/ (sin x) x)))
     (if (<= y 1.15e+103) (/ (* (sinh y) x) x) (/ (* y (* (sin x) t_0)) x)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 0.0064) {
		tmp = y * (t_0 * (sin(x) / x));
	} else if (y <= 1.15e+103) {
		tmp = (sinh(y) * x) / x;
	} else {
		tmp = (y * (sin(x) * t_0)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if (y <= 0.0064d0) then
        tmp = y * (t_0 * (sin(x) / x))
    else if (y <= 1.15d+103) then
        tmp = (sinh(y) * x) / x
    else
        tmp = (y * (sin(x) * t_0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 0.0064) {
		tmp = y * (t_0 * (Math.sin(x) / x));
	} else if (y <= 1.15e+103) {
		tmp = (Math.sinh(y) * x) / x;
	} else {
		tmp = (y * (Math.sin(x) * t_0)) / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if y <= 0.0064:
		tmp = y * (t_0 * (math.sin(x) / x))
	elif y <= 1.15e+103:
		tmp = (math.sinh(y) * x) / x
	else:
		tmp = (y * (math.sin(x) * t_0)) / x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (y <= 0.0064)
		tmp = Float64(y * Float64(t_0 * Float64(sin(x) / x)));
	elseif (y <= 1.15e+103)
		tmp = Float64(Float64(sinh(y) * x) / x);
	else
		tmp = Float64(Float64(y * Float64(sin(x) * t_0)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (y <= 0.0064)
		tmp = y * (t_0 * (sin(x) / x));
	elseif (y <= 1.15e+103)
		tmp = (sinh(y) * x) / x;
	else
		tmp = (y * (sin(x) * t_0)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.0064], N[(y * N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+103], N[(N[(N[Sinh[y], $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision], N[(N[(y * N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.00640000000000000031

   1. Initial program 83.7%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 87.1%

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

   if 0.00640000000000000031 < y < 1.15000000000000004e103

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{sinh.f64}\left(y\right)\right), x\right) \]
   4. Step-by-step derivation

   5. Simplified 81.8%

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

   if 1.15000000000000004e103 < y

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 87.9%

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

      1. *-commutative N/A

         \[\leadsto \left(\left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right) \cdot \color{blue}{y} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right) \cdot \sin x}{x} \cdot y \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(\left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right) \cdot \sin x\right) \cdot y}{\color{blue}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right) \cdot \sin x\right) \cdot y\right), \color{blue}{x}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right) \cdot \sin x\right), y\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sin x \cdot \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right), y\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin x, \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right), y\right), x\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right), y\right), x\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)\right)\right), y\right), x\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot y\right)\right)\right)\right), y\right), x\right) \]

      11. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), y\right), x\right) \]

   9. Applied egg-rr 100.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0064:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sinh y \cdot x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)\\ \mathbf{if}\;y \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* (+ 1.0 (* 0.16666666666666666 (* y y))) (/ (sin x) x)))))
   (if (<= y 9.2e-7) t_0 (if (<= y 2e+121) (sinh y) t_0))))

C

double code(double x, double y) {
	double t_0 = y * ((1.0 + (0.16666666666666666 * (y * y))) * (sin(x) / x));
	double tmp;
	if (y <= 9.2e-7) {
		tmp = t_0;
	} else if (y <= 2e+121) {
		tmp = sinh(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * ((1.0d0 + (0.16666666666666666d0 * (y * y))) * (sin(x) / x))
    if (y <= 9.2d-7) then
        tmp = t_0
    else if (y <= 2d+121) then
        tmp = sinh(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * ((1.0 + (0.16666666666666666 * (y * y))) * (Math.sin(x) / x));
	double tmp;
	if (y <= 9.2e-7) {
		tmp = t_0;
	} else if (y <= 2e+121) {
		tmp = Math.sinh(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * ((1.0 + (0.16666666666666666 * (y * y))) * (math.sin(x) / x))
	tmp = 0
	if y <= 9.2e-7:
		tmp = t_0
	elif y <= 2e+121:
		tmp = math.sinh(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(sin(x) / x)))
	tmp = 0.0
	if (y <= 9.2e-7)
		tmp = t_0;
	elseif (y <= 2e+121)
		tmp = sinh(y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * ((1.0 + (0.16666666666666666 * (y * y))) * (sin(x) / x));
	tmp = 0.0;
	if (y <= 9.2e-7)
		tmp = t_0;
	elseif (y <= 2e+121)
		tmp = sinh(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 9.2e-7], t$95$0, If[LessEqual[y, 2e+121], N[Sinh[y], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 9.1999999999999998e-7 or 2.00000000000000007e121 < y

   1. Initial program 86.1%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 88.2%

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

   if 9.1999999999999998e-7 < y < 2.00000000000000007e121

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.6%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

         \[\leadsto \sinh y \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 78.6%

         \[\leadsto \mathsf{sinh.f64}\left(y\right) \]

   9. Applied egg-rr 78.6%

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

Alternative 8: 69.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9.2e-7) (/ y (/ x (sin x))) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-7) {
		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 (y <= 9.2d-7) 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 (y <= 9.2e-7) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.2e-7:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.2e-7)
		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 (y <= 9.2e-7)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.2e-7], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.1999999999999998e-7

   1. Initial program 83.5%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

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

      1. *-commutative N/A

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

      2. associate-/r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sinh y, \color{blue}{\left(\frac{x}{\sin x}\right)}\right) \]

      4. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \left(\frac{\color{blue}{x}}{\sin x}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \mathsf{/.f64}\left(x, \color{blue}{\sin x}\right)\right) \]

      6. sin-lowering-sin.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{y}, \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 69.8%

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

   if 9.1999999999999998e-7 < y

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 79.4%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

         \[\leadsto \sinh y \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 79.4%

         \[\leadsto \mathsf{sinh.f64}\left(y\right) \]

   9. Applied egg-rr 79.4%

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

Alternative 9: 73.9% 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 87.6%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
6. Step-by-step derivation

7. Simplified 75.6%

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

Alternative 10: 54.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.8)
   (/ y (+ 1.0 (* x (* x 0.16666666666666666))))
   (* x (/ (* y (* y (* y (* y (* y 0.008333333333333333))))) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.8) {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = x * ((y * (y * (y * (y * (y * 0.008333333333333333))))) / 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 <= 4.8d0) then
        tmp = y / (1.0d0 + (x * (x * 0.16666666666666666d0)))
    else
        tmp = x * ((y * (y * (y * (y * (y * 0.008333333333333333d0))))) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.8) {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = x * ((y * (y * (y * (y * (y * 0.008333333333333333))))) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.8:
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)))
	else:
		tmp = x * ((y * (y * (y * (y * (y * 0.008333333333333333))))) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.8)
		tmp = Float64(y / Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666))));
	else
		tmp = Float64(x * Float64(Float64(y * Float64(y * Float64(y * Float64(y * Float64(y * 0.008333333333333333))))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.8)
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	else
		tmp = x * ((y * (y * (y * (y * (y * 0.008333333333333333))))) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.8], N[(y / N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.79999999999999982

   1. Initial program 83.8%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

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

      1. *-commutative N/A

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

      2. associate-/r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sinh y, \color{blue}{\left(\frac{x}{\sin x}\right)}\right) \]

      4. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \left(\frac{\color{blue}{x}}{\sin x}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \mathsf{/.f64}\left(x, \color{blue}{\sin x}\right)\right) \]

      6. sin-lowering-sin.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{y}, \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 69.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(1 + \frac{1}{6} \cdot {x}^{2}\right)}\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       6. *-lowering-*.f64 53.7%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   12. Simplified 53.7%

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

   if 4.79999999999999982 < y

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.3%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}, x\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      11. *-lowering-*.f64 70.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 70.5%

       \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}}{x} \]

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right), x\right)\right) \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot 2\right)}\right)\right), x\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)\right), x\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \frac{1}{120}\right)\right)\right)\right), x\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\left(y \cdot y\right) \cdot \frac{1}{120}\right)\right)\right)\right), x\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \frac{1}{120}\right)\right)\right)\right)\right), x\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} \cdot y\right)\right)\right)\right)\right), x\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot y\right)\right)\right)\right)\right), x\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{120}\right)\right)\right)\right)\right), x\right)\right) \]

       15. *-lowering-*.f64 70.5%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right)\right), x\right)\right) \]

   13. Simplified 70.5%

       \[\leadsto x \cdot \frac{y \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\right)}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 69.0% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}{x} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  x
  (/
   (*
    y
    (+
     1.0
     (* y (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333))))))
   x)))

C

double code(double x, double y) {
	return x * ((y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * ((y * (1.0d0 + (y * (y * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0)))))) / x)
end function

Java

public static double code(double x, double y) {
	return x * ((y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) / x);
}

Python

def code(x, y):
	return x * ((y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) / x)

Julia

function code(x, y)
	return Float64(x * Float64(Float64(y * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333)))))) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) / x);
end

Wolfram

code[x_, y_] := N[(x * N[(N[(y * N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.6%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
6. Step-by-step derivation

7. Simplified 75.6%

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

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}, x\right)\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

   11. *-lowering-*.f64 72.3%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

10. Simplified 72.3%

    \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}}{x} \]
11. Add Preprocessing

Alternative 12: 66.7% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6500000:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot 0.16666666666666666\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6500000.0)
   (* x (/ (* y (+ 1.0 (* y (* y 0.16666666666666666)))) x))
   (/ (* x (* (* y y) (* y 0.16666666666666666))) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6500000.0) {
		tmp = x * ((y * (1.0 + (y * (y * 0.16666666666666666)))) / x);
	} else {
		tmp = (x * ((y * y) * (y * 0.16666666666666666))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 6500000.0d0) then
        tmp = x * ((y * (1.0d0 + (y * (y * 0.16666666666666666d0)))) / x)
    else
        tmp = (x * ((y * y) * (y * 0.16666666666666666d0))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6500000.0) {
		tmp = x * ((y * (1.0 + (y * (y * 0.16666666666666666)))) / x);
	} else {
		tmp = (x * ((y * y) * (y * 0.16666666666666666))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6500000.0:
		tmp = x * ((y * (1.0 + (y * (y * 0.16666666666666666)))) / x)
	else:
		tmp = (x * ((y * y) * (y * 0.16666666666666666))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6500000.0)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666)))) / x));
	else
		tmp = Float64(Float64(x * Float64(Float64(y * y) * Float64(y * 0.16666666666666666))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6500000.0)
		tmp = x * ((y * (1.0 + (y * (y * 0.16666666666666666)))) / x);
	else
		tmp = (x * ((y * y) * (y * 0.16666666666666666))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6500000.0], N[(x * N[(N[(y * N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5e6

   1. Initial program 83.7%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 82.4%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}, x\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \frac{1}{6}\right)\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), x\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \frac{1}{6}\right)\right)\right)\right), x\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)\right)\right), x\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot y\right)\right)\right)\right), x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), x\right)\right) \]

      9. *-lowering-*.f64 77.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 77.3%

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

   if 6.5e6 < x

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 53.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2}}{x} + \frac{1}{x}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{\frac{1}{6} \cdot {y}^{2}}{x} + \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{{y}^{2} \cdot \frac{1}{6}}{x} + \frac{1}{x}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \frac{\frac{1}{6}}{x} + \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \frac{\frac{1}{6} \cdot 1}{x} + \frac{1}{x}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{x} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{{y}^{2}} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot y\right), \color{blue}{y}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right), y\right), y\right)\right)\right)\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{6} \cdot 1}{x}\right), y\right), y\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{6}}{x}\right), y\right), y\right)\right)\right)\right) \]

      17. /-lowering-/.f64 40.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{6}, x\right), y\right), y\right)\right)\right)\right) \]

   10. Simplified 40.0%

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

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot \frac{{y}^{3}}{x}\right)}\right) \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \frac{\left(y \cdot y\right) \cdot y}{x}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot y}{x}\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(\frac{{y}^{2}}{x} \cdot \color{blue}{y}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \frac{{y}^{2}}{x}\right) \cdot \color{blue}{y}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{{y}^{2}}{x}\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot \frac{{y}^{2}}{x}\right)}\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{\frac{1}{6} \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{6} \cdot {y}^{2}\right), \color{blue}{x}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2}\right)\right), x\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot y\right)\right), x\right)\right)\right) \]

       11. *-lowering-*.f64 42.6%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right), x\right)\right)\right) \]

   13. Simplified 42.6%

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

       1. *-commutative N/A

          \[\leadsto \left(y \cdot \frac{\frac{1}{6} \cdot \left(y \cdot y\right)}{x}\right) \cdot \color{blue}{x} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{y \cdot \left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)}{x} \cdot x \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot x}{\color{blue}{x}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot x\right), \color{blue}{x}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)\right), x\right), x\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(y \cdot \frac{1}{6}\right) \cdot \left(y \cdot y\right)\right), x\right), x\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{1}{6} \cdot y\right) \cdot \left(y \cdot y\right)\right), x\right), x\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{6} \cdot y\right), \left(y \cdot y\right)\right), x\right), x\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot \frac{1}{6}\right), \left(y \cdot y\right)\right), x\right), x\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \frac{1}{6}\right), \left(y \cdot y\right)\right), x\right), x\right) \]

       11. *-lowering-*.f64 49.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \frac{1}{6}\right), \mathsf{*.f64}\left(y, y\right)\right), x\right), x\right) \]

   15. Applied egg-rr 49.9%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6500000:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot 0.16666666666666666\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.7% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6500000:\\ \;\;\;\;x \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot 0.16666666666666666\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6500000.0)
   (* x (* (+ 1.0 (* 0.16666666666666666 (* y y))) (/ y x)))
   (/ (* x (* (* y y) (* y 0.16666666666666666))) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6500000.0) {
		tmp = x * ((1.0 + (0.16666666666666666 * (y * y))) * (y / x));
	} else {
		tmp = (x * ((y * y) * (y * 0.16666666666666666))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 6500000.0d0) then
        tmp = x * ((1.0d0 + (0.16666666666666666d0 * (y * y))) * (y / x))
    else
        tmp = (x * ((y * y) * (y * 0.16666666666666666d0))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6500000.0) {
		tmp = x * ((1.0 + (0.16666666666666666 * (y * y))) * (y / x));
	} else {
		tmp = (x * ((y * y) * (y * 0.16666666666666666))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6500000.0:
		tmp = x * ((1.0 + (0.16666666666666666 * (y * y))) * (y / x))
	else:
		tmp = (x * ((y * y) * (y * 0.16666666666666666))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6500000.0)
		tmp = Float64(x * Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(y / x)));
	else
		tmp = Float64(Float64(x * Float64(Float64(y * y) * Float64(y * 0.16666666666666666))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6500000.0)
		tmp = x * ((1.0 + (0.16666666666666666 * (y * y))) * (y / x));
	else
		tmp = (x * ((y * y) * (y * 0.16666666666666666))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6500000.0], N[(x * N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5e6

   1. Initial program 83.7%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 82.4%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2}}{x} + \frac{1}{x}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{\frac{1}{6} \cdot {y}^{2}}{x} + \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{{y}^{2} \cdot \frac{1}{6}}{x} + \frac{1}{x}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \frac{\frac{1}{6}}{x} + \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \frac{\frac{1}{6} \cdot 1}{x} + \frac{1}{x}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{x} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{{y}^{2}} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot y\right), \color{blue}{y}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right), y\right), y\right)\right)\right)\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{6} \cdot 1}{x}\right), y\right), y\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{6}}{x}\right), y\right), y\right)\right)\right)\right) \]

      17. /-lowering-/.f64 77.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{6}, x\right), y\right), y\right)\right)\right)\right) \]

   10. Simplified 77.2%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)}{x}\right)}\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot y}{x}\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{y}{x}}\right)\right) \]

       3. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{y \cdot 1}{x}\right)\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \left(y \cdot \color{blue}{\frac{1}{x}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(y \cdot \frac{1}{x}\right)}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right), \left(\color{blue}{y} \cdot \frac{1}{x}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2}\right)\right)\right), \left(y \cdot \frac{1}{x}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot y\right)\right)\right), \left(y \cdot \frac{1}{x}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \left(y \cdot \frac{1}{x}\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \left(\frac{y \cdot 1}{\color{blue}{x}}\right)\right)\right) \]

       11. *-rgt-identity N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \left(\frac{y}{x}\right)\right)\right) \]

       12. /-lowering-/.f64 77.3%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 77.3%

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

   if 6.5e6 < x

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 53.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2}}{x} + \frac{1}{x}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{\frac{1}{6} \cdot {y}^{2}}{x} + \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{{y}^{2} \cdot \frac{1}{6}}{x} + \frac{1}{x}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \frac{\frac{1}{6}}{x} + \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \frac{\frac{1}{6} \cdot 1}{x} + \frac{1}{x}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{x} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{{y}^{2}} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot y\right), \color{blue}{y}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right), y\right), y\right)\right)\right)\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{6} \cdot 1}{x}\right), y\right), y\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{6}}{x}\right), y\right), y\right)\right)\right)\right) \]

      17. /-lowering-/.f64 40.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{6}, x\right), y\right), y\right)\right)\right)\right) \]

   10. Simplified 40.0%

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

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot \frac{{y}^{3}}{x}\right)}\right) \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \frac{\left(y \cdot y\right) \cdot y}{x}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot y}{x}\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(\frac{{y}^{2}}{x} \cdot \color{blue}{y}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \frac{{y}^{2}}{x}\right) \cdot \color{blue}{y}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{{y}^{2}}{x}\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot \frac{{y}^{2}}{x}\right)}\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{\frac{1}{6} \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{6} \cdot {y}^{2}\right), \color{blue}{x}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2}\right)\right), x\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot y\right)\right), x\right)\right)\right) \]

       11. *-lowering-*.f64 42.6%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right), x\right)\right)\right) \]

   13. Simplified 42.6%

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

       1. *-commutative N/A

          \[\leadsto \left(y \cdot \frac{\frac{1}{6} \cdot \left(y \cdot y\right)}{x}\right) \cdot \color{blue}{x} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{y \cdot \left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)}{x} \cdot x \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot x}{\color{blue}{x}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot x\right), \color{blue}{x}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)\right), x\right), x\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(y \cdot \frac{1}{6}\right) \cdot \left(y \cdot y\right)\right), x\right), x\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{1}{6} \cdot y\right) \cdot \left(y \cdot y\right)\right), x\right), x\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{6} \cdot y\right), \left(y \cdot y\right)\right), x\right), x\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot \frac{1}{6}\right), \left(y \cdot y\right)\right), x\right), x\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \frac{1}{6}\right), \left(y \cdot y\right)\right), x\right), x\right) \]

       11. *-lowering-*.f64 49.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \frac{1}{6}\right), \mathsf{*.f64}\left(y, y\right)\right), x\right), x\right) \]

   15. Applied egg-rr 49.9%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6500000:\\ \;\;\;\;x \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot 0.16666666666666666\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.2% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.16666666666666666 \cdot \left(y \cdot y\right)}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.8)
   (/ y (+ 1.0 (* x (* x 0.16666666666666666))))
   (* x (* y (/ (* 0.16666666666666666 (* y y)) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.8) {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = x * (y * ((0.16666666666666666 * (y * y)) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.8d0) then
        tmp = y / (1.0d0 + (x * (x * 0.16666666666666666d0)))
    else
        tmp = x * (y * ((0.16666666666666666d0 * (y * y)) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.8) {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = x * (y * ((0.16666666666666666 * (y * y)) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.8:
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)))
	else:
		tmp = x * (y * ((0.16666666666666666 * (y * y)) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.8)
		tmp = Float64(y / Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666))));
	else
		tmp = Float64(x * Float64(y * Float64(Float64(0.16666666666666666 * Float64(y * y)) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.8)
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	else
		tmp = x * (y * ((0.16666666666666666 * (y * y)) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.8], N[(y / N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7999999999999998

   1. Initial program 83.8%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

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

      1. *-commutative N/A

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

      2. associate-/r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sinh y, \color{blue}{\left(\frac{x}{\sin x}\right)}\right) \]

      4. sinh-lowering-sinh.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \left(\frac{\color{blue}{x}}{\sin x}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \mathsf{/.f64}\left(x, \color{blue}{\sin x}\right)\right) \]

      6. sin-lowering-sin.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{y}, \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 69.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(1 + \frac{1}{6} \cdot {x}^{2}\right)}\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       6. *-lowering-*.f64 53.7%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   12. Simplified 53.7%

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

   if 3.7999999999999998 < y

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.3%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2}}{x} + \frac{1}{x}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{\frac{1}{6} \cdot {y}^{2}}{x} + \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{{y}^{2} \cdot \frac{1}{6}}{x} + \frac{1}{x}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \frac{\frac{1}{6}}{x} + \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \frac{\frac{1}{6} \cdot 1}{x} + \frac{1}{x}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{x} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{{y}^{2}} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot y\right), \color{blue}{y}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right), y\right), y\right)\right)\right)\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{6} \cdot 1}{x}\right), y\right), y\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{6}}{x}\right), y\right), y\right)\right)\right)\right) \]

      17. /-lowering-/.f64 64.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{6}, x\right), y\right), y\right)\right)\right)\right) \]

   10. Simplified 64.2%

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

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot \frac{{y}^{3}}{x}\right)}\right) \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \frac{\left(y \cdot y\right) \cdot y}{x}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot y}{x}\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(\frac{{y}^{2}}{x} \cdot \color{blue}{y}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \frac{{y}^{2}}{x}\right) \cdot \color{blue}{y}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{{y}^{2}}{x}\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot \frac{{y}^{2}}{x}\right)}\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{\frac{1}{6} \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{6} \cdot {y}^{2}\right), \color{blue}{x}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2}\right)\right), x\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot y\right)\right), x\right)\right)\right) \]

       11. *-lowering-*.f64 64.2%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right), x\right)\right)\right) \]

   13. Simplified 64.2%

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

Alternative 15: 57.3% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+107}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.6e+107)
   (* x (/ 1.0 (/ x y)))
   (* y (* 0.16666666666666666 (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.6e+107) {
		tmp = x * (1.0 / (x / y));
	} else {
		tmp = y * (0.16666666666666666 * (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.6d+107) then
        tmp = x * (1.0d0 / (x / y))
    else
        tmp = y * (0.16666666666666666d0 * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.6e+107) {
		tmp = x * (1.0 / (x / y));
	} else {
		tmp = y * (0.16666666666666666 * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.6e+107:
		tmp = x * (1.0 / (x / y))
	else:
		tmp = y * (0.16666666666666666 * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.6e+107)
		tmp = Float64(x * Float64(1.0 / Float64(x / y)));
	else
		tmp = Float64(y * Float64(0.16666666666666666 * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.6e+107)
		tmp = x * (1.0 / (x / y));
	else
		tmp = y * (0.16666666666666666 * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.6e+107], N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.6000000000000001e107

   1. Initial program 85.4%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{y}, x\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 54.2%

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

       1. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{x}{y}}}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

       3. /-lowering-/.f64 55.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   12. Applied egg-rr 55.4%

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

   if 4.6000000000000001e107 < y

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2}}{x} + \frac{1}{x}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{\frac{1}{6} \cdot {y}^{2}}{x} + \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{{y}^{2} \cdot \frac{1}{6}}{x} + \frac{1}{x}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \frac{\frac{1}{6}}{x} + \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \frac{\frac{1}{6} \cdot 1}{x} + \frac{1}{x}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{x} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{{y}^{2}} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot y\right), \color{blue}{y}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right), y\right), y\right)\right)\right)\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{6} \cdot 1}{x}\right), y\right), y\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{6}}{x}\right), y\right), y\right)\right)\right)\right) \]

      17. /-lowering-/.f64 78.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{6}, x\right), y\right), y\right)\right)\right)\right) \]

   10. Simplified 78.9%

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

   11. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3}} \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{1}{6} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot \left({y}^{2} \cdot y\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{y} \]

       4. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

       8. *-lowering-*.f64 78.9%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   13. Simplified 78.9%

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

Alternative 16: 51.3% accurate, 29.3× 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(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 87.6%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
6. Step-by-step derivation

7. Simplified 75.6%

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

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{y}, x\right)\right) \]
9. Step-by-step derivation

10. Simplified 53.9%

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

    1. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{x}{y}}}\right)\right) \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

    3. /-lowering-/.f64 54.8%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

12. Applied egg-rr 54.8%

    \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{x}{y}}} \]
13. Add Preprocessing

Alternative 17: 50.8% 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 87.6%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
6. Step-by-step derivation

7. Simplified 75.6%

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

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{y}, x\right)\right) \]
9. Step-by-step derivation

10. Simplified 53.9%

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

Alternative 18: 28.0% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 87.6%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
6. Step-by-step derivation

7. Simplified 75.6%

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

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{y} \]
9. Step-by-step derivation

10. Simplified 29.6%

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

Developer Target 1: 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 2024158 
(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))