Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.9% → 99.9%

Time: 7.4s

Alternatives: 6

Speedup: 1.0×

• 49.2% 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: 88.9% 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 86.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{y}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.4e+154) (* (sin x) (/ y x)) (sqrt (pow y 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.4e+154) {
		tmp = sin(x) * (y / x);
	} else {
		tmp = sqrt(pow(y, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.4d+154) then
        tmp = sin(x) * (y / x)
    else
        tmp = sqrt((y ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.4e+154) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sqrt(Math.pow(y, 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.4e+154:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sqrt(math.pow(y, 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.4e+154)
		tmp = Float64(sin(x) * Float64(y / x));
	else
		tmp = sqrt((y ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.4e+154)
		tmp = sin(x) * (y / x);
	else
		tmp = sqrt((y ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.4e+154], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Power[y, 2.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4e154

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. add-sqr-sqrt 38.8%

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

      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. Taylor expanded in y around 0 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-*r/ 66.6%

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

   7. Simplified 66.6%

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

   if 1.4e154 < y

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 6.3%

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

   6. Taylor expanded in x around 0 21.1%

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

      1. add-sqr-sqrt 21.1%

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

      2. sqrt-unprod 93.8%

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

      3. pow2 93.8%

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

      4. *-commutative 93.8%

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

      5. associate-/l* 93.8%

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

   8. Applied egg-rr 93.8%

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

      1. *-inverses 93.8%

         \[\leadsto \sqrt{{\left(y \cdot \color{blue}{1}\right)}^{2}} \]

      2. *-rgt-identity 93.8%

         \[\leadsto \sqrt{{\color{blue}{y}}^{2}} \]

   10. Simplified 93.8%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{y}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+42}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.35e+42) (* y (/ (sin x) x)) (/ (/ y x) (/ 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.35e+42) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = (y / x) / (1.0 / 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 <= 1.35d+42) then
        tmp = y * (sin(x) / x)
    else
        tmp = (y / x) / (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.35e+42) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = (y / x) / (1.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.35e+42:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = (y / x) / (1.0 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.35e+42)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(Float64(y / x) / Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.35e+42)
		tmp = y * (sin(x) / x);
	else
		tmp = (y / x) / (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.35e+42], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e42

   1. Initial program 82.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 45.3%

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

      1. associate-/l* 62.9%

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

   7. Simplified 62.9%

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

   if 1.35e42 < 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 41.1%

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

      3. times-frac 41.1%

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

   4. Applied egg-rr 41.1%

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

   5. Taylor expanded in y around 0 5.1%

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

      1. *-commutative 5.1%

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

      2. associate-*r/ 51.9%

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

   7. Simplified 51.9%

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

      1. associate-*r/ 5.1%

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

      2. *-commutative 5.1%

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

      3. associate-*l/ 51.9%

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

      4. associate-/r/ 5.1%

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

      5. div-inv 5.1%

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

      6. associate-/r* 51.9%

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

   9. Applied egg-rr 51.9%

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

   10. Taylor expanded in x around 0 51.8%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+42}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

   1. *-commutative 86.1%

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

   2. add-sqr-sqrt 39.0%

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

   3. times-frac 45.2%

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

4. Applied egg-rr 45.2%

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

5. Taylor expanded in y around 0 36.5%

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

   1. *-commutative 36.5%

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

   2. associate-*r/ 66.3%

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

7. Simplified 66.3%

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

8. Final simplification 66.3%

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

Alternative 5: 50.4% 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 86.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 36.5%

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

6. Taylor expanded in x around 0 18.7%

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

   1. associate-/l* 52.6%

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

   2. *-commutative 52.6%

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

8. Applied egg-rr 52.6%

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

9. Final simplification 52.6%

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

Alternative 6: 28.4% 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 86.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 36.5%

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

   1. associate-/l* 50.2%

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

7. Simplified 50.2%

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

8. Taylor expanded in x around 0 26.7%

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

9. Final simplification 26.7%

   \[\leadsto y \]
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 2024055 
(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))