Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

Alternatives: 9

Speedup: 1.0×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 55.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+16}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+117} \lor \neg \left(y \leq 3.1 \cdot 10^{+215}\right):\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(x \cdot \left(0.125 + x \cdot -0.05\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(x \cdot 0.125 - 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1e+16)
   (sin x)
   (if (or (<= y 3e+117) (not (<= y 3.1e+215)))
     (*
      x
      (+
       1.0
       (* (pow x 2.0) (- (* x (+ 0.125 (* x -0.05))) 0.16666666666666666))))
     (* x (+ 1.0 (* (pow x 2.0) (- (* x 0.125) 0.16666666666666666)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e+16) {
		tmp = sin(x);
	} else if ((y <= 3e+117) || !(y <= 3.1e+215)) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((x * (0.125 + (x * -0.05))) - 0.16666666666666666)));
	} else {
		tmp = x * (1.0 + (pow(x, 2.0) * ((x * 0.125) - 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1d+16) then
        tmp = sin(x)
    else if ((y <= 3d+117) .or. (.not. (y <= 3.1d+215))) then
        tmp = x * (1.0d0 + ((x ** 2.0d0) * ((x * (0.125d0 + (x * (-0.05d0)))) - 0.16666666666666666d0)))
    else
        tmp = x * (1.0d0 + ((x ** 2.0d0) * ((x * 0.125d0) - 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1e+16) {
		tmp = Math.sin(x);
	} else if ((y <= 3e+117) || !(y <= 3.1e+215)) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((x * (0.125 + (x * -0.05))) - 0.16666666666666666)));
	} else {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((x * 0.125) - 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e+16:
		tmp = math.sin(x)
	elif (y <= 3e+117) or not (y <= 3.1e+215):
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((x * (0.125 + (x * -0.05))) - 0.16666666666666666)))
	else:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((x * 0.125) - 0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e+16)
		tmp = sin(x);
	elseif ((y <= 3e+117) || !(y <= 3.1e+215))
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(x * Float64(0.125 + Float64(x * -0.05))) - 0.16666666666666666))));
	else
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(x * 0.125) - 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e+16)
		tmp = sin(x);
	elseif ((y <= 3e+117) || ~((y <= 3.1e+215)))
		tmp = x * (1.0 + ((x ^ 2.0) * ((x * (0.125 + (x * -0.05))) - 0.16666666666666666)));
	else
		tmp = x * (1.0 + ((x ^ 2.0) * ((x * 0.125) - 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e+16], N[Sin[x], $MachinePrecision], If[Or[LessEqual[y, 3e+117], N[Not[LessEqual[y, 3.1e+215]], $MachinePrecision]], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(x * N[(0.125 + N[(x * -0.05), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(x * 0.125), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 66.7%

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

   if 1e16 < y < 3e117 or 3.0999999999999999e215 < y

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\sin x} - 1}\right) \]
   6. Step-by-step derivation

      1. expm1-define 2.6%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   7. Simplified 2.6%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   8. Taylor expanded in x around 0 6.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(1 + 0.5 \cdot x\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 6.3%

         \[\leadsto \mathsf{log1p}\left(x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right)\right) \]

   10. Simplified 6.3%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(1 + x \cdot 0.5\right)}\right) \]

   11. Taylor expanded in x around 0 32.2%

       \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(x \cdot \left(0.125 + -0.05 \cdot x\right) - 0.16666666666666666\right)\right)} \]

   if 3e117 < y < 3.0999999999999999e215

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.1%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\sin x} - 1}\right) \]
   6. Step-by-step derivation

      1. expm1-define 2.5%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   7. Simplified 2.5%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   8. Taylor expanded in x around 0 5.6%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(1 + 0.5 \cdot x\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 5.6%

         \[\leadsto \mathsf{log1p}\left(x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right)\right) \]

   10. Simplified 5.6%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(1 + x \cdot 0.5\right)}\right) \]

   11. Taylor expanded in x around 0 12.9%

       \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.125 \cdot x - 0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+16}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+117} \lor \neg \left(y \leq 3.1 \cdot 10^{+215}\right):\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(x \cdot \left(0.125 + x \cdot -0.05\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(x \cdot 0.125 - 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+16}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 2.46 \cdot 10^{+117} \lor \neg \left(y \leq 7.5 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{x \cdot \left(y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(x \cdot 0.125 - 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1e+16)
   (sin x)
   (if (or (<= y 2.46e+117) (not (<= y 7.5e+216)))
     (/ (* x (+ y (* -0.16666666666666666 (* y (pow x 2.0))))) y)
     (* x (+ 1.0 (* (pow x 2.0) (- (* x 0.125) 0.16666666666666666)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e+16) {
		tmp = sin(x);
	} else if ((y <= 2.46e+117) || !(y <= 7.5e+216)) {
		tmp = (x * (y + (-0.16666666666666666 * (y * pow(x, 2.0))))) / y;
	} else {
		tmp = x * (1.0 + (pow(x, 2.0) * ((x * 0.125) - 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1d+16) then
        tmp = sin(x)
    else if ((y <= 2.46d+117) .or. (.not. (y <= 7.5d+216))) then
        tmp = (x * (y + ((-0.16666666666666666d0) * (y * (x ** 2.0d0))))) / y
    else
        tmp = x * (1.0d0 + ((x ** 2.0d0) * ((x * 0.125d0) - 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1e+16) {
		tmp = Math.sin(x);
	} else if ((y <= 2.46e+117) || !(y <= 7.5e+216)) {
		tmp = (x * (y + (-0.16666666666666666 * (y * Math.pow(x, 2.0))))) / y;
	} else {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((x * 0.125) - 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e+16:
		tmp = math.sin(x)
	elif (y <= 2.46e+117) or not (y <= 7.5e+216):
		tmp = (x * (y + (-0.16666666666666666 * (y * math.pow(x, 2.0))))) / y
	else:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((x * 0.125) - 0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e+16)
		tmp = sin(x);
	elseif ((y <= 2.46e+117) || !(y <= 7.5e+216))
		tmp = Float64(Float64(x * Float64(y + Float64(-0.16666666666666666 * Float64(y * (x ^ 2.0))))) / y);
	else
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(x * 0.125) - 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e+16)
		tmp = sin(x);
	elseif ((y <= 2.46e+117) || ~((y <= 7.5e+216)))
		tmp = (x * (y + (-0.16666666666666666 * (y * (x ^ 2.0))))) / y;
	else
		tmp = x * (1.0 + ((x ^ 2.0) * ((x * 0.125) - 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e+16], N[Sin[x], $MachinePrecision], If[Or[LessEqual[y, 2.46e+117], N[Not[LessEqual[y, 7.5e+216]], $MachinePrecision]], N[(N[(x * N[(y + N[(-0.16666666666666666 * N[(y * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(x * 0.125), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 66.7%

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

   if 1e16 < y < 2.46000000000000002e117 or 7.4999999999999994e216 < y

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 2.6%

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

   8. Taylor expanded in x around 0 32.2%

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

   if 2.46000000000000002e117 < y < 7.4999999999999994e216

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.1%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\sin x} - 1}\right) \]
   6. Step-by-step derivation

      1. expm1-define 2.5%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   7. Simplified 2.5%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   8. Taylor expanded in x around 0 5.6%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(1 + 0.5 \cdot x\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 5.6%

         \[\leadsto \mathsf{log1p}\left(x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right)\right) \]

   10. Simplified 5.6%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(1 + x \cdot 0.5\right)}\right) \]

   11. Taylor expanded in x around 0 12.9%

       \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.125 \cdot x - 0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+16}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 2.46 \cdot 10^{+117} \lor \neg \left(y \leq 7.5 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{x \cdot \left(y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(x \cdot 0.125 - 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+16}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+120} \lor \neg \left(y \leq 1.65 \cdot 10^{+216}\right):\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(x \cdot 0.125 - 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1e+16)
   (sin x)
   (if (or (<= y 6.2e+120) (not (<= y 1.65e+216)))
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (* x (+ 1.0 (* (pow x 2.0) (- (* x 0.125) 0.16666666666666666)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e+16) {
		tmp = sin(x);
	} else if ((y <= 6.2e+120) || !(y <= 1.65e+216)) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = x * (1.0 + (pow(x, 2.0) * ((x * 0.125) - 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1d+16) then
        tmp = sin(x)
    else if ((y <= 6.2d+120) .or. (.not. (y <= 1.65d+216))) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = x * (1.0d0 + ((x ** 2.0d0) * ((x * 0.125d0) - 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1e+16) {
		tmp = Math.sin(x);
	} else if ((y <= 6.2e+120) || !(y <= 1.65e+216)) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((x * 0.125) - 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e+16:
		tmp = math.sin(x)
	elif (y <= 6.2e+120) or not (y <= 1.65e+216):
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((x * 0.125) - 0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e+16)
		tmp = sin(x);
	elseif ((y <= 6.2e+120) || !(y <= 1.65e+216))
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(x * 0.125) - 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e+16)
		tmp = sin(x);
	elseif ((y <= 6.2e+120) || ~((y <= 1.65e+216)))
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = x * (1.0 + ((x ^ 2.0) * ((x * 0.125) - 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e+16], N[Sin[x], $MachinePrecision], If[Or[LessEqual[y, 6.2e+120], N[Not[LessEqual[y, 1.65e+216]], $MachinePrecision]], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(x * 0.125), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 66.7%

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

   if 1e16 < y < 6.19999999999999947e120 or 1.65e216 < y

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\sin x} - 1}\right) \]
   6. Step-by-step derivation

      1. expm1-define 2.6%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   7. Simplified 2.6%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   8. Taylor expanded in x around 0 22.9%

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

      1. distribute-rgt-in 22.9%

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

      2. *-lft-identity 22.9%

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

      3. associate-*l* 22.9%

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

      4. pow-plus 22.9%

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

      5. metadata-eval 22.9%

         \[\leadsto x + -0.16666666666666666 \cdot {x}^{\color{blue}{3}} \]

   10. Simplified 22.9%

       \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]

   if 6.19999999999999947e120 < y < 1.65e216

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.1%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\sin x} - 1}\right) \]
   6. Step-by-step derivation

      1. expm1-define 2.5%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   7. Simplified 2.5%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   8. Taylor expanded in x around 0 5.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(1 + 0.5 \cdot x\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 5.7%

         \[\leadsto \mathsf{log1p}\left(x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right)\right) \]

   10. Simplified 5.7%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(1 + x \cdot 0.5\right)}\right) \]

   11. Taylor expanded in x around 0 13.3%

       \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.125 \cdot x - 0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+16}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+120} \lor \neg \left(y \leq 1.65 \cdot 10^{+216}\right):\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(x \cdot 0.125 - 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+16}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e+16) (sin x) (+ x (* -0.16666666666666666 (pow x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.05e+16) {
		tmp = sin(x);
	} else {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.05d+16) then
        tmp = sin(x)
    else
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.05e+16) {
		tmp = Math.sin(x);
	} else {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.05e+16:
		tmp = math.sin(x)
	else:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.05e+16)
		tmp = sin(x);
	else
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.05e+16)
		tmp = sin(x);
	else
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.05e+16], N[Sin[x], $MachinePrecision], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 66.7%

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

   if 1.05e16 < y

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.2%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\sin x} - 1}\right) \]
   6. Step-by-step derivation

      1. expm1-define 2.6%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   7. Simplified 2.6%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   8. Taylor expanded in x around 0 18.0%

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

      1. distribute-rgt-in 18.0%

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

      2. *-lft-identity 18.0%

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

      3. associate-*l* 18.0%

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

      4. pow-plus 18.0%

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

      5. metadata-eval 18.0%

         \[\leadsto x + -0.16666666666666666 \cdot {x}^{\color{blue}{3}} \]

   10. Simplified 18.0%

       \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 54.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+16}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1e+16) (sin x) (* -0.16666666666666666 (pow x 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e+16) {
		tmp = sin(x);
	} else {
		tmp = -0.16666666666666666 * pow(x, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1d+16) then
        tmp = sin(x)
    else
        tmp = (-0.16666666666666666d0) * (x ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1e+16) {
		tmp = Math.sin(x);
	} else {
		tmp = -0.16666666666666666 * Math.pow(x, 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e+16:
		tmp = math.sin(x)
	else:
		tmp = -0.16666666666666666 * math.pow(x, 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e+16)
		tmp = sin(x);
	else
		tmp = Float64(-0.16666666666666666 * (x ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e+16)
		tmp = sin(x);
	else
		tmp = -0.16666666666666666 * (x ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e+16], N[Sin[x], $MachinePrecision], N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 66.7%

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

   if 1e16 < y

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.2%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\sin x} - 1}\right) \]
   6. Step-by-step derivation

      1. expm1-define 2.6%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   7. Simplified 2.6%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   8. Taylor expanded in x around 0 18.0%

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

   9. Taylor expanded in x around inf 17.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 53.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{+59}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.95e+59) (sin x) (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.95e+59) {
		tmp = sin(x);
	} else {
		tmp = (x * 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 <= 1.95d+59) then
        tmp = sin(x)
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.95e+59) {
		tmp = Math.sin(x);
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.95e+59:
		tmp = math.sin(x)
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.95e+59)
		tmp = sin(x);
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.95e+59)
		tmp = sin(x);
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.95e+59], N[Sin[x], $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.95000000000000011e59

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 64.4%

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

   if 1.95000000000000011e59 < y

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 2.6%

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

   8. Taylor expanded in x around 0 11.4%

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

Alternative 8: 29.4% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+42}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1e+42) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e+42) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1d+42) then
        tmp = x
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1e+42) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1e+42:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e+42)
		tmp = x;
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1e+42)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e+42], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000004e42

   1. Initial program 100.0%

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

      1. log1p-expm1-u 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 20.8%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\sin x} - 1}\right) \]
   6. Step-by-step derivation

      1. expm1-define 49.1%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   7. Simplified 49.1%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

   8. Taylor expanded in x around 0 32.3%

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

   if 1.00000000000000004e42 < x

   1. Initial program 100.0%

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

      1. add-log-exp 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. log-prod 99.4%

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

      4. metadata-eval 99.4%

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

      5. add-log-exp 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in y around 0 50.7%

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

   8. Taylor expanded in x around 0 7.6%

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

Alternative 9: 26.2% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

   1. log1p-expm1-u 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in y around 0 28.6%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\sin x} - 1}\right) \]
6. Step-by-step derivation

   1. expm1-define 49.5%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

7. Simplified 49.5%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin x\right)}\right) \]

8. Taylor expanded in x around 0 24.3%

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

Reproduce

herbie shell --seed 2024100 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))