Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.8% → 99.9%

Time: 8.4s

Alternatives: 5

Speedup: 1.0×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 69.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 5.0) (* y (/ (sin x) x)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 5.0) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sinh(y) <= 5.0d0) then
        tmp = y * (sin(x) / x)
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= 5.0) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 5.0:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 5.0)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 5.0)
		tmp = y * (sin(x) / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 5.0], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 5

   1. Initial program 84.1%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 55.0%

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

      1. associate-/l* 70.8%

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

   7. Simplified 70.8%

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

   if 5 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-sqr-sqrt 44.4%

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

      3. times-frac 44.4%

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

   4. Applied egg-rr 44.4%

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

      1. frac-times 44.4%

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

      2. add-sqr-sqrt 100.0%

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

      3. associate-/l* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/r/ 100.0%

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

      6. div-inv 100.0%

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

      7. associate-/r* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 77.8%

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

      1. remove-double-div 77.8%

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

      2. sinh-def 77.8%

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

      3. div-sub 77.8%

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

   9. Applied egg-rr 77.8%

      \[\leadsto \color{blue}{\frac{e^{y}}{2} - \frac{e^{-y}}{2}} \]
   10. Step-by-step derivation

       1. div-sub 77.8%

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

       2. sinh-def 77.8%

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

   11. Simplified 77.8%

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

4. Final simplification 72.5%

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

Alternative 3: 62.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 5 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 5e-45) (* x (/ y x)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 5e-45) {
		tmp = 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) <= 5d-45) then
        tmp = 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) <= 5e-45) {
		tmp = x * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 5e-45:
		tmp = x * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 5e-45)
		tmp = Float64(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) <= 5e-45)
		tmp = x * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 5e-45], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 4.99999999999999976e-45

   1. Initial program 83.3%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 53.2%

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

   6. Taylor expanded in x around 0 22.8%

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

      1. associate-/l* 57.2%

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

      2. *-commutative 57.2%

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

   8. Applied egg-rr 57.2%

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

   if 4.99999999999999976e-45 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-sqr-sqrt 45.8%

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

      3. times-frac 45.8%

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

   4. Applied egg-rr 45.8%

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

      1. frac-times 45.8%

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

      2. add-sqr-sqrt 100.0%

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

      3. associate-/l* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/r/ 100.0%

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

      6. div-inv 99.9%

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

      7. associate-/r* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 79.2%

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

      1. remove-double-div 79.2%

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

      2. sinh-def 69.2%

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

      3. div-sub 69.2%

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

   9. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\frac{e^{y}}{2} - \frac{e^{-y}}{2}} \]
   10. Step-by-step derivation

       1. div-sub 69.2%

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

       2. sinh-def 79.2%

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

   11. Simplified 79.2%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 5 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.9% 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 88.0%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around 0 42.6%

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

6. Taylor expanded in x around 0 23.1%

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

   1. associate-/l* 51.5%

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

   2. *-commutative 51.5%

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

8. Applied egg-rr 51.5%

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

9. Final simplification 51.5%

   \[\leadsto x \cdot \frac{y}{x} \]
10. Add Preprocessing

Alternative 5: 28.3% 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 88.0%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around 0 42.6%

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

   1. associate-/l* 54.5%

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

7. Simplified 54.5%

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

8. Taylor expanded in x around 0 29.2%

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

9. Final simplification 29.2%

   \[\leadsto y \]
10. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024059 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

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

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