Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 5

Speedup: 1.0×

• 49.4% 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: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 1.0000000002:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.0000000002) (sin x) (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.0000000002) {
		tmp = sin(x);
	} else {
		tmp = x * 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 = sinh(y) / y
    if (t_0 <= 1.0000000002d0) then
        tmp = sin(x)
    else
        tmp = x * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(y) / y;
	double tmp;
	if (t_0 <= 1.0000000002) {
		tmp = Math.sin(x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if t_0 <= 1.0000000002:
		tmp = math.sin(x)
	else:
		tmp = x * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (t_0 <= 1.0000000002)
		tmp = sin(x);
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	tmp = 0.0;
	if (t_0 <= 1.0000000002)
		tmp = sin(x);
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, 1.0000000002], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1.0000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.7%

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

   if 1.0000000002 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. clear-num 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 77.2%

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

      1. associate-*r/ 77.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot y}{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}}} \]

      2. *-commutative 77.2%

         \[\leadsto \frac{1}{\frac{2 \cdot y}{\color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot x}}} \]

      3. associate-/r* 77.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{2 \cdot y}{e^{y} - \frac{1}{e^{y}}}}{x}}} \]

      4. associate-*r/ 77.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot \frac{y}{e^{y} - \frac{1}{e^{y}}}}}{x}} \]

      5. *-commutative 77.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{e^{y} - \frac{1}{e^{y}}} \cdot 2}}{x}} \]

      6. associate-/r/ 77.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{\frac{e^{y} - \frac{1}{e^{y}}}{2}}}}{x}} \]

      7. rec-exp 77.2%

         \[\leadsto \frac{1}{\frac{\frac{y}{\frac{e^{y} - \color{blue}{e^{-y}}}{2}}}{x}} \]

      8. sinh-def 77.4%

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

   7. Simplified 77.4%

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

      1. associate-/r/ 77.4%

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

      2. clear-num 77.4%

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

   9. Applied egg-rr 77.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 1.0000000002:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 400000:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 400000.0) (sin x) (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 400000.0) {
		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 <= 400000.0d0) 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 <= 400000.0) {
		tmp = Math.sin(x);
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 400000.0:
		tmp = math.sin(x)
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 400000.0)
		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 <= 400000.0)
		tmp = sin(x);
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 400000.0], N[Sin[x], $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4e5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 63.5%

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

   if 4e5 < 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 2.5%

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

   8. Taylor expanded in x around 0 8.4%

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

Alternative 4: 29.0% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.5e-6) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-6) {
		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 (y <= 1.5d-6) 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 (y <= 1.5e-6) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.5e-6:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.5e-6)
		tmp = x;
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.5e-6)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.5e-6], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5e-6

   1. Initial program 100.0%

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

      1. associate-*r/ 88.5%

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

      2. clear-num 88.3%

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

   4. Applied egg-rr 88.3%

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

   5. Taylor expanded in x around 0 28.5%

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

      1. associate-*r/ 28.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot y}{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}}} \]

      2. *-commutative 28.5%

         \[\leadsto \frac{1}{\frac{2 \cdot y}{\color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot x}}} \]

      3. associate-/r* 28.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{2 \cdot y}{e^{y} - \frac{1}{e^{y}}}}{x}}} \]

      4. associate-*r/ 28.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot \frac{y}{e^{y} - \frac{1}{e^{y}}}}}{x}} \]

      5. *-commutative 28.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{e^{y} - \frac{1}{e^{y}}} \cdot 2}}{x}} \]

      6. associate-/r/ 28.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{\frac{e^{y} - \frac{1}{e^{y}}}{2}}}}{x}} \]

      7. rec-exp 28.5%

         \[\leadsto \frac{1}{\frac{\frac{y}{\frac{e^{y} - \color{blue}{e^{-y}}}{2}}}{x}} \]

      8. sinh-def 59.1%

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

   7. Simplified 59.1%

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

   8. Taylor expanded in y around 0 34.8%

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

   if 1.5e-6 < 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 97.2%

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

      2. *-un-lft-identity 97.2%

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

      3. log-prod 97.2%

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

      4. metadata-eval 97.2%

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

      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 4.1%

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

   8. Taylor expanded in x around 0 9.3%

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

Alternative 5: 25.8% 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. associate-*r/ 91.5%

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

   2. clear-num 91.4%

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

4. Applied egg-rr 91.4%

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

5. Taylor expanded in x around 0 43.6%

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

   1. associate-*r/ 43.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot y}{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}}} \]

   2. *-commutative 43.6%

      \[\leadsto \frac{1}{\frac{2 \cdot y}{\color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot x}}} \]

   3. associate-/r* 43.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{2 \cdot y}{e^{y} - \frac{1}{e^{y}}}}{x}}} \]

   4. associate-*r/ 43.6%

      \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot \frac{y}{e^{y} - \frac{1}{e^{y}}}}}{x}} \]

   5. *-commutative 43.6%

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{e^{y} - \frac{1}{e^{y}}} \cdot 2}}{x}} \]

   6. associate-/r/ 43.6%

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{\frac{e^{y} - \frac{1}{e^{y}}}{2}}}}{x}} \]

   7. rec-exp 43.7%

      \[\leadsto \frac{1}{\frac{\frac{y}{\frac{e^{y} - \color{blue}{e^{-y}}}{2}}}{x}} \]

   8. sinh-def 66.3%

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

7. Simplified 66.3%

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

8. Taylor expanded in y around 0 26.6%

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

Reproduce

herbie shell --seed 2024108 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))