Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 7

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 3.7274190193991457e+149
  2. y = 3.3402622168268575e+125

• 49.6% 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: 87.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) / y) <= 1.0) {
		tmp = sin(x);
	} else {
		tmp = x / (y * (1.0 / 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) / y) <= 1.0d0) then
        tmp = sin(x)
    else
        tmp = x / (y * (1.0d0 / sinh(y)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   if 1 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 68.6%

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

      1. clear-num 68.6%

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

      2. un-div-inv 68.6%

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

   5. Applied egg-rr 68.6%

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

      1. clear-num 68.6%

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

      2. associate-/r/ 68.6%

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

   7. Applied egg-rr 68.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 1:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{1}{\sinh y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) / y) <= 1.0) {
		tmp = sin(x);
	} else {
		tmp = x / (y / 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) / y) <= 1.0d0) then
        tmp = sin(x)
    else
        tmp = x / (y / sinh(y))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   if 1 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 68.6%

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

      1. clear-num 68.6%

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

      2. un-div-inv 68.6%

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

   5. Applied egg-rr 68.6%

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

Alternative 4: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 1:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.0) (sin x) (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		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 <= 1.0:
		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 <= 1.0)
		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 <= 1.0)
		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, 1.0], 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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   if 1 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 68.6%

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

Alternative 5: 54.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+14}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.5e+14) (sin x) (* -0.16666666666666666 (pow x 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e+14) {
		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 <= 8.5d+14) 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 <= 8.5e+14) {
		tmp = Math.sin(x);
	} else {
		tmp = -0.16666666666666666 * Math.pow(x, 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.5e+14:
		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 <= 8.5e+14)
		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 <= 8.5e+14)
		tmp = sin(x);
	else
		tmp = -0.16666666666666666 * (x ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e+14], N[Sin[x], $MachinePrecision], N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5e14

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 68.1%

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

   if 8.5e14 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 2.7%

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

   4. Taylor expanded in x around 0 22.3%

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

      1. distribute-rgt-in 22.3%

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

      2. *-lft-identity 22.3%

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

      3. associate-*l* 22.3%

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

      4. pow-plus 22.3%

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

      5. metadata-eval 22.3%

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

   6. Simplified 22.3%

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

   7. Taylor expanded in x around inf 21.9%

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

Alternative 6: 51.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return sin(x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Sin[x], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 48.4%

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

Alternative 7: 26.8% 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. Taylor expanded in x around 0 61.8%

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

4. Taylor expanded in y around 0 26.7%

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

Reproduce

herbie shell --seed 2024165 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))