Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.7% → 99.8%

Time: 9.0s

Alternatives: 9

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 6.377889675011589e+80
  2. y = 3.781101626837031e+304

• 49.7% 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: 88.7% 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: 99.8% 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 89.1%

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

Alternative 2: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 200000:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 200000.0) (* y (/ (sin x) x)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 200000.0) {
		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) <= 200000.0d0) 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) <= 200000.0) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 200000.0:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 200000.0)
		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) <= 200000.0)
		tmp = y * (sin(x) / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 200000.0], 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) < 2e5

   1. Initial program 85.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 50.4%

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

      1. associate-/l* 65.3%

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

   7. Simplified 65.3%

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

   if 2e5 < (sinh.f64 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 x around 0 72.1%

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

      1. clear-num 72.1%

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

      2. un-div-inv 72.1%

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

   7. Applied egg-rr 72.1%

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

      1. associate-/r/ 72.1%

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

      2. *-inverses 72.1%

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

      3. *-lft-identity 72.1%

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

   9. Simplified 72.1%

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

Alternative 3: 62.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2e-69) (/ x (/ x y)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2e-69) {
		tmp = x / (x / y);
	} 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) <= 2d-69) then
        tmp = x / (x / y)
    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) <= 2e-69) {
		tmp = x / (x / y);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 2e-69:
		tmp = x / (x / y)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 2e-69)
		tmp = Float64(x / Float64(x / y));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 2e-69)
		tmp = x / (x / y);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 2e-69], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 1.9999999999999999e-69

   1. Initial program 83.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 72.0%

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

   6. Taylor expanded in x around 0 57.9%

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

      1. clear-num 59.4%

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

      2. un-div-inv 58.4%

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

   8. Applied egg-rr 58.4%

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

   if 1.9999999999999999e-69 < (sinh.f64 y)

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 x around 0 67.0%

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

      1. clear-num 67.0%

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

      2. un-div-inv 67.0%

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

   7. Applied egg-rr 67.0%

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

      1. associate-/r/ 67.0%

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

      2. *-inverses 67.0%

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

      3. *-lft-identity 67.0%

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

   9. Simplified 67.0%

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

Alternative 4: 74.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 x around 0 71.6%

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

Alternative 5: 54.7% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.06 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.06e-43)
   (/ x (/ x y))
   (* (+ x (* -0.16666666666666666 (* x (* x x)))) (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.06e-43) {
		tmp = x / (x / y);
	} else {
		tmp = (x + (-0.16666666666666666 * (x * (x * x)))) * (y / 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 <= 1.06d-43) then
        tmp = x / (x / y)
    else
        tmp = (x + ((-0.16666666666666666d0) * (x * (x * x)))) * (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.06e-43) {
		tmp = x / (x / y);
	} else {
		tmp = (x + (-0.16666666666666666 * (x * (x * x)))) * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.06e-43:
		tmp = x / (x / y)
	else:
		tmp = (x + (-0.16666666666666666 * (x * (x * x)))) * (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.06e-43)
		tmp = Float64(x / Float64(x / y));
	else
		tmp = Float64(Float64(x + Float64(-0.16666666666666666 * Float64(x * Float64(x * x)))) * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.06e-43)
		tmp = x / (x / y);
	else
		tmp = (x + (-0.16666666666666666 * (x * (x * x)))) * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.06e-43], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05999999999999994e-43

   1. Initial program 84.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 72.4%

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

   6. Taylor expanded in x around 0 57.3%

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

      1. clear-num 58.8%

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

      2. un-div-inv 57.8%

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

   8. Applied egg-rr 57.8%

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

   if 1.05999999999999994e-43 < y

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 40.5%

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

   6. Taylor expanded in x around 0 48.0%

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

      1. distribute-rgt-in 48.0%

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

      2. *-lft-identity 48.0%

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

      3. associate-*l* 48.0%

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

      4. unpow2 48.0%

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

      5. unpow3 48.0%

         \[\leadsto \left(x + -0.16666666666666666 \cdot \color{blue}{{x}^{3}}\right) \cdot \frac{y}{x} \]

   8. Simplified 48.0%

      \[\leadsto \color{blue}{\left(x + -0.16666666666666666 \cdot {x}^{3}\right)} \cdot \frac{y}{x} \]
   9. Step-by-step derivation

      1. unpow3 48.0%

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

      2. pow2 48.0%

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

   10. Applied egg-rr 48.0%

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

       1. unpow2 48.0%

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

   12. Applied egg-rr 48.0%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.06 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.4% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+216}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.8e+138)
   (* x (/ 1.0 (/ x y)))
   (if (<= y 5.6e+216) (* -0.16666666666666666 (* y (* x x))) (* x (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.8e+138) {
		tmp = x * (1.0 / (x / y));
	} else if (y <= 5.6e+216) {
		tmp = -0.16666666666666666 * (y * (x * x));
	} else {
		tmp = x * (y / 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 <= 6.8d+138) then
        tmp = x * (1.0d0 / (x / y))
    else if (y <= 5.6d+216) then
        tmp = (-0.16666666666666666d0) * (y * (x * x))
    else
        tmp = x * (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.8e+138) {
		tmp = x * (1.0 / (x / y));
	} else if (y <= 5.6e+216) {
		tmp = -0.16666666666666666 * (y * (x * x));
	} else {
		tmp = x * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.8e+138:
		tmp = x * (1.0 / (x / y))
	elif y <= 5.6e+216:
		tmp = -0.16666666666666666 * (y * (x * x))
	else:
		tmp = x * (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.8e+138)
		tmp = Float64(x * Float64(1.0 / Float64(x / y)));
	elseif (y <= 5.6e+216)
		tmp = Float64(-0.16666666666666666 * Float64(y * Float64(x * x)));
	else
		tmp = Float64(x * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.8e+138)
		tmp = x * (1.0 / (x / y));
	elseif (y <= 5.6e+216)
		tmp = -0.16666666666666666 * (y * (x * x));
	else
		tmp = x * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.8e+138], N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.6e+216], N[(-0.16666666666666666 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.80000000000000022e138

   1. Initial program 86.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 67.3%

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

      1. clear-num 66.7%

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

      2. associate-/r/ 67.3%

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

   7. Applied egg-rr 67.3%

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

   8. Taylor expanded in x around 0 52.0%

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

      1. associate-/r/ 53.3%

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

   10. Applied egg-rr 53.3%

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

   if 6.80000000000000022e138 < y < 5.59999999999999963e216

   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 23.5%

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

   6. Taylor expanded in x around 0 42.7%

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

      1. distribute-rgt-in 42.7%

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

      2. *-lft-identity 42.7%

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

      3. associate-*l* 42.7%

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

      4. unpow2 42.7%

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

      5. unpow3 42.7%

         \[\leadsto \left(x + -0.16666666666666666 \cdot \color{blue}{{x}^{3}}\right) \cdot \frac{y}{x} \]

   8. Simplified 42.7%

      \[\leadsto \color{blue}{\left(x + -0.16666666666666666 \cdot {x}^{3}\right)} \cdot \frac{y}{x} \]

   9. Taylor expanded in x around inf 21.3%

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

       1. unpow2 42.7%

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

   11. Applied egg-rr 21.3%

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

   if 5.59999999999999963e216 < 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 55.6%

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

   6. Taylor expanded in x around 0 54.2%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+216}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.6% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{+138}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+216}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.8e+138)
   (/ x (/ x y))
   (if (<= y 6.5e+216) (* -0.16666666666666666 (* y (* x x))) (* x (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.8e+138) {
		tmp = x / (x / y);
	} else if (y <= 6.5e+216) {
		tmp = -0.16666666666666666 * (y * (x * x));
	} else {
		tmp = x * (y / 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 <= 7.8d+138) then
        tmp = x / (x / y)
    else if (y <= 6.5d+216) then
        tmp = (-0.16666666666666666d0) * (y * (x * x))
    else
        tmp = x * (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.8e+138) {
		tmp = x / (x / y);
	} else if (y <= 6.5e+216) {
		tmp = -0.16666666666666666 * (y * (x * x));
	} else {
		tmp = x * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.8e+138:
		tmp = x / (x / y)
	elif y <= 6.5e+216:
		tmp = -0.16666666666666666 * (y * (x * x))
	else:
		tmp = x * (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.8e+138)
		tmp = Float64(x / Float64(x / y));
	elseif (y <= 6.5e+216)
		tmp = Float64(-0.16666666666666666 * Float64(y * Float64(x * x)));
	else
		tmp = Float64(x * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.8e+138)
		tmp = x / (x / y);
	elseif (y <= 6.5e+216)
		tmp = -0.16666666666666666 * (y * (x * x));
	else
		tmp = x * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.8e+138], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+216], N[(-0.16666666666666666 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.7999999999999996e138

   1. Initial program 86.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 67.3%

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

   6. Taylor expanded in x around 0 52.1%

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

      1. clear-num 53.3%

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

      2. un-div-inv 52.0%

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

   8. Applied egg-rr 52.0%

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

   if 7.7999999999999996e138 < y < 6.50000000000000029e216

   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 23.5%

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

   6. Taylor expanded in x around 0 42.7%

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

      1. distribute-rgt-in 42.7%

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

      2. *-lft-identity 42.7%

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

      3. associate-*l* 42.7%

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

      4. unpow2 42.7%

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

      5. unpow3 42.7%

         \[\leadsto \left(x + -0.16666666666666666 \cdot \color{blue}{{x}^{3}}\right) \cdot \frac{y}{x} \]

   8. Simplified 42.7%

      \[\leadsto \color{blue}{\left(x + -0.16666666666666666 \cdot {x}^{3}\right)} \cdot \frac{y}{x} \]

   9. Taylor expanded in x around inf 21.3%

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

       1. unpow2 42.7%

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

   11. Applied egg-rr 21.3%

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

   if 6.50000000000000029e216 < 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 55.6%

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

   6. Taylor expanded in x around 0 54.2%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{+138}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+216}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 89.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 62.3%

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

6. Taylor expanded in x around 0 49.3%

   \[\leadsto \color{blue}{x} \cdot \frac{y}{x} \]
7. Add Preprocessing

Alternative 9: 28.1% 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 89.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 x around 0 71.6%

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

6. Taylor expanded in y around 0 25.4%

   \[\leadsto \color{blue}{y} \]
7. Add Preprocessing

Developer Target 1: 99.8% 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 2024181 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

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