Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.9% → 99.8%

Time: 9.5s

Alternatives: 12

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: 88.9% 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: 99.8% 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 92.3%

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

   1. associate-*r/ 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. Final simplification 99.9%

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

Alternative 2: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)\\ \mathbf{if}\;y \leq -490:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 720:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.35 \cdot 10^{+248}:\\ \;\;\;\;y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (log1p (expm1 y))))
   (if (<= y -490.0)
     t_0
     (if (<= y 720.0)
       (/ y (/ x (sin x)))
       (if (<= y 2.1e+226)
         t_0
         (if (<= y 4.35e+248)
           (* y (* x (* x -0.16666666666666666)))
           (sqrt (* y y))))))))

C

double code(double x, double y) {
	double t_0 = log1p(expm1(y));
	double tmp;
	if (y <= -490.0) {
		tmp = t_0;
	} else if (y <= 720.0) {
		tmp = y / (x / sin(x));
	} else if (y <= 2.1e+226) {
		tmp = t_0;
	} else if (y <= 4.35e+248) {
		tmp = y * (x * (x * -0.16666666666666666));
	} else {
		tmp = sqrt((y * y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.log1p(Math.expm1(y));
	double tmp;
	if (y <= -490.0) {
		tmp = t_0;
	} else if (y <= 720.0) {
		tmp = y / (x / Math.sin(x));
	} else if (y <= 2.1e+226) {
		tmp = t_0;
	} else if (y <= 4.35e+248) {
		tmp = y * (x * (x * -0.16666666666666666));
	} else {
		tmp = Math.sqrt((y * y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.log1p(math.expm1(y))
	tmp = 0
	if y <= -490.0:
		tmp = t_0
	elif y <= 720.0:
		tmp = y / (x / math.sin(x))
	elif y <= 2.1e+226:
		tmp = t_0
	elif y <= 4.35e+248:
		tmp = y * (x * (x * -0.16666666666666666))
	else:
		tmp = math.sqrt((y * y))
	return tmp

Julia

function code(x, y)
	t_0 = log1p(expm1(y))
	tmp = 0.0
	if (y <= -490.0)
		tmp = t_0;
	elseif (y <= 720.0)
		tmp = Float64(y / Float64(x / sin(x)));
	elseif (y <= 2.1e+226)
		tmp = t_0;
	elseif (y <= 4.35e+248)
		tmp = Float64(y * Float64(x * Float64(x * -0.16666666666666666)));
	else
		tmp = sqrt(Float64(y * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Log[1 + N[(Exp[y] - 1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -490.0], t$95$0, If[LessEqual[y, 720.0], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+226], t$95$0, If[LessEqual[y, 4.35e+248], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -490 or 720 < y < 2.09999999999999993e226

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 4.4%

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

   5. Taylor expanded in x around 0 14.9%

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

      1. *-commutative 14.9%

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

   7. Simplified 14.9%

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

      1. div-inv 14.9%

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

      2. associate-*l* 4.2%

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

      3. div-inv 4.2%

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

      4. *-inverses 4.2%

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

      5. *-commutative 4.2%

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

      6. *-un-lft-identity 4.2%

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

      7. log1p-expm1-u 80.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]

   9. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]

   if -490 < y < 720

   1. Initial program 84.3%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 83.8%

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

      1. associate-/l* 99.4%

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

   6. Simplified 99.4%

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

   if 2.09999999999999993e226 < y < 4.34999999999999995e248

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 3.4%

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

      1. associate-/l* 3.4%

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

      2. associate-/r/ 3.4%

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

   6. Simplified 3.4%

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

   7. Taylor expanded in x around 0 85.7%

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

      1. +-commutative 85.7%

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

      2. remove-double-neg 85.7%

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

      3. unsub-neg 85.7%

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

      4. associate-*r* 85.7%

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

      5. *-commutative 85.7%

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

      6. fma-neg 85.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.16666666666666666 \cdot {x}^{2}, -\left(-y\right)\right)} \]

      7. unpow2 85.7%

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

      8. associate-*r* 85.7%

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

      9. remove-double-neg 85.7%

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

   9. Simplified 85.7%

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

   10. Taylor expanded in x around inf 85.7%

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

       1. associate-*r* 85.7%

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

       2. *-commutative 85.7%

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

       3. unpow2 85.7%

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

       4. associate-*r* 85.7%

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

   12. Simplified 85.7%

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

   if 4.34999999999999995e248 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 8.0%

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

   5. Taylor expanded in x around 0 43.7%

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

      1. *-commutative 43.7%

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

   7. Simplified 43.7%

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

      1. div-inv 43.7%

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

      2. associate-*l* 9.5%

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

      3. div-inv 9.5%

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

      4. *-inverses 9.5%

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

      5. *-commutative 9.5%

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

      6. *-un-lft-identity 9.5%

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

      7. add-sqr-sqrt 9.5%

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

      8. sqrt-unprod 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\sqrt{y \cdot y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -490:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)\\ \mathbf{elif}\;y \leq 720:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+226}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)\\ \mathbf{elif}\;y \leq 4.35 \cdot 10^{+248}:\\ \;\;\;\;y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \]

Alternative 3: 61.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{if}\;y \leq -270:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.175:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 6.0 (/ (/ y x) x))))
   (if (<= y -270.0)
     t_0
     (if (<= y 0.175)
       (/ y (+ 1.0 (* x (* x 0.16666666666666666))))
       (if (<= y 1.42e+135) t_0 (sqrt (* y y)))))))

C

double code(double x, double y) {
	double t_0 = 6.0 * ((y / x) / x);
	double tmp;
	if (y <= -270.0) {
		tmp = t_0;
	} else if (y <= 0.175) {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	} else if (y <= 1.42e+135) {
		tmp = t_0;
	} else {
		tmp = sqrt((y * y));
	}
	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 = 6.0d0 * ((y / x) / x)
    if (y <= (-270.0d0)) then
        tmp = t_0
    else if (y <= 0.175d0) then
        tmp = y / (1.0d0 + (x * (x * 0.16666666666666666d0)))
    else if (y <= 1.42d+135) then
        tmp = t_0
    else
        tmp = sqrt((y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 6.0 * ((y / x) / x);
	double tmp;
	if (y <= -270.0) {
		tmp = t_0;
	} else if (y <= 0.175) {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	} else if (y <= 1.42e+135) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((y * y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 6.0 * ((y / x) / x)
	tmp = 0
	if y <= -270.0:
		tmp = t_0
	elif y <= 0.175:
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)))
	elif y <= 1.42e+135:
		tmp = t_0
	else:
		tmp = math.sqrt((y * y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(6.0 * Float64(Float64(y / x) / x))
	tmp = 0.0
	if (y <= -270.0)
		tmp = t_0;
	elseif (y <= 0.175)
		tmp = Float64(y / Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666))));
	elseif (y <= 1.42e+135)
		tmp = t_0;
	else
		tmp = sqrt(Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 6.0 * ((y / x) / x);
	tmp = 0.0;
	if (y <= -270.0)
		tmp = t_0;
	elseif (y <= 0.175)
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	elseif (y <= 1.42e+135)
		tmp = t_0;
	else
		tmp = sqrt((y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(6.0 * N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -270.0], t$95$0, If[LessEqual[y, 0.175], N[(y / N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.42e+135], t$95$0, N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -270 or 0.17499999999999999 < y < 1.41999999999999998e135

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 4.7%

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

      1. associate-/l* 4.7%

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

   6. Simplified 4.7%

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

   7. Taylor expanded in x around 0 3.9%

      \[\leadsto \frac{y}{\color{blue}{1 + 0.16666666666666666 \cdot {x}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 3.9%

         \[\leadsto \frac{y}{1 + \color{blue}{{x}^{2} \cdot 0.16666666666666666}} \]

      2. unpow2 3.9%

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

   9. Simplified 3.9%

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

   10. Taylor expanded in x around inf 41.1%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 41.1%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

       2. associate-/r* 41.2%

          \[\leadsto 6 \cdot \color{blue}{\frac{\frac{y}{x}}{x}} \]

   12. Simplified 41.2%

       \[\leadsto \color{blue}{6 \cdot \frac{\frac{y}{x}}{x}} \]

   if -270 < y < 0.17499999999999999

   1. Initial program 84.2%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 84.2%

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

      1. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 82.3%

      \[\leadsto \frac{y}{\color{blue}{1 + 0.16666666666666666 \cdot {x}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 82.3%

         \[\leadsto \frac{y}{1 + \color{blue}{{x}^{2} \cdot 0.16666666666666666}} \]

      2. unpow2 82.3%

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

   9. Simplified 82.3%

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

   10. Taylor expanded in x around 0 82.3%

       \[\leadsto \frac{y}{1 + \color{blue}{0.16666666666666666 \cdot {x}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 82.3%

          \[\leadsto \frac{y}{1 + \color{blue}{{x}^{2} \cdot 0.16666666666666666}} \]

       2. unpow2 82.3%

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

       3. associate-*r* 82.3%

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

   12. Simplified 82.3%

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

   if 1.41999999999999998e135 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 5.1%

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

   5. Taylor expanded in x around 0 26.3%

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

      1. *-commutative 26.3%

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

   7. Simplified 26.3%

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

      1. div-inv 26.3%

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

      2. associate-*l* 5.0%

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

      3. div-inv 5.0%

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

      4. *-inverses 5.0%

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

      5. *-commutative 5.0%

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

      6. *-un-lft-identity 5.0%

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

      7. add-sqr-sqrt 5.0%

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

      8. sqrt-unprod 61.9%

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

   9. Applied egg-rr 61.9%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -270:\\ \;\;\;\;6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 0.175:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+135}:\\ \;\;\;\;6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \]

Alternative 4: 72.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{if}\;y \leq -430:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 450:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 6.0 (/ (/ y x) x))))
   (if (<= y -430.0)
     t_0
     (if (<= y 450.0)
       (* (sin x) (/ y x))
       (if (<= y 1.42e+135) t_0 (sqrt (* y y)))))))

C

double code(double x, double y) {
	double t_0 = 6.0 * ((y / x) / x);
	double tmp;
	if (y <= -430.0) {
		tmp = t_0;
	} else if (y <= 450.0) {
		tmp = sin(x) * (y / x);
	} else if (y <= 1.42e+135) {
		tmp = t_0;
	} else {
		tmp = sqrt((y * y));
	}
	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 = 6.0d0 * ((y / x) / x)
    if (y <= (-430.0d0)) then
        tmp = t_0
    else if (y <= 450.0d0) then
        tmp = sin(x) * (y / x)
    else if (y <= 1.42d+135) then
        tmp = t_0
    else
        tmp = sqrt((y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 6.0 * ((y / x) / x);
	double tmp;
	if (y <= -430.0) {
		tmp = t_0;
	} else if (y <= 450.0) {
		tmp = Math.sin(x) * (y / x);
	} else if (y <= 1.42e+135) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((y * y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 6.0 * ((y / x) / x)
	tmp = 0
	if y <= -430.0:
		tmp = t_0
	elif y <= 450.0:
		tmp = math.sin(x) * (y / x)
	elif y <= 1.42e+135:
		tmp = t_0
	else:
		tmp = math.sqrt((y * y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(6.0 * Float64(Float64(y / x) / x))
	tmp = 0.0
	if (y <= -430.0)
		tmp = t_0;
	elseif (y <= 450.0)
		tmp = Float64(sin(x) * Float64(y / x));
	elseif (y <= 1.42e+135)
		tmp = t_0;
	else
		tmp = sqrt(Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 6.0 * ((y / x) / x);
	tmp = 0.0;
	if (y <= -430.0)
		tmp = t_0;
	elseif (y <= 450.0)
		tmp = sin(x) * (y / x);
	elseif (y <= 1.42e+135)
		tmp = t_0;
	else
		tmp = sqrt((y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(6.0 * N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -430.0], t$95$0, If[LessEqual[y, 450.0], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.42e+135], t$95$0, N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -430 or 450 < y < 1.41999999999999998e135

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 4.5%

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

      1. associate-/l* 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in x around 0 3.8%

      \[\leadsto \frac{y}{\color{blue}{1 + 0.16666666666666666 \cdot {x}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 3.8%

         \[\leadsto \frac{y}{1 + \color{blue}{{x}^{2} \cdot 0.16666666666666666}} \]

      2. unpow2 3.8%

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

   9. Simplified 3.8%

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

   10. Taylor expanded in x around inf 41.5%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 41.5%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

       2. associate-/r* 41.5%

          \[\leadsto 6 \cdot \color{blue}{\frac{\frac{y}{x}}{x}} \]

   12. Simplified 41.5%

       \[\leadsto \color{blue}{6 \cdot \frac{\frac{y}{x}}{x}} \]

   if -430 < y < 450

   1. Initial program 84.3%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 83.8%

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

      1. associate-/l* 99.4%

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

      2. associate-/r/ 99.2%

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

   6. Simplified 99.2%

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

   if 1.41999999999999998e135 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 5.1%

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

   5. Taylor expanded in x around 0 26.3%

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

      1. *-commutative 26.3%

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

   7. Simplified 26.3%

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

      1. div-inv 26.3%

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

      2. associate-*l* 5.0%

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

      3. div-inv 5.0%

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

      4. *-inverses 5.0%

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

      5. *-commutative 5.0%

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

      6. *-un-lft-identity 5.0%

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

      7. add-sqr-sqrt 5.0%

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

      8. sqrt-unprod 61.9%

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

   9. Applied egg-rr 61.9%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -430:\\ \;\;\;\;6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 450:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+135}:\\ \;\;\;\;6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \]

Alternative 5: 72.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{if}\;y \leq -660:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 330:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 6.0 (/ (/ y x) x))))
   (if (<= y -660.0)
     t_0
     (if (<= y 330.0)
       (/ y (/ x (sin x)))
       (if (<= y 1.42e+135) t_0 (sqrt (* y y)))))))

C

double code(double x, double y) {
	double t_0 = 6.0 * ((y / x) / x);
	double tmp;
	if (y <= -660.0) {
		tmp = t_0;
	} else if (y <= 330.0) {
		tmp = y / (x / sin(x));
	} else if (y <= 1.42e+135) {
		tmp = t_0;
	} else {
		tmp = sqrt((y * y));
	}
	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 = 6.0d0 * ((y / x) / x)
    if (y <= (-660.0d0)) then
        tmp = t_0
    else if (y <= 330.0d0) then
        tmp = y / (x / sin(x))
    else if (y <= 1.42d+135) then
        tmp = t_0
    else
        tmp = sqrt((y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 6.0 * ((y / x) / x);
	double tmp;
	if (y <= -660.0) {
		tmp = t_0;
	} else if (y <= 330.0) {
		tmp = y / (x / Math.sin(x));
	} else if (y <= 1.42e+135) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((y * y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 6.0 * ((y / x) / x)
	tmp = 0
	if y <= -660.0:
		tmp = t_0
	elif y <= 330.0:
		tmp = y / (x / math.sin(x))
	elif y <= 1.42e+135:
		tmp = t_0
	else:
		tmp = math.sqrt((y * y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(6.0 * Float64(Float64(y / x) / x))
	tmp = 0.0
	if (y <= -660.0)
		tmp = t_0;
	elseif (y <= 330.0)
		tmp = Float64(y / Float64(x / sin(x)));
	elseif (y <= 1.42e+135)
		tmp = t_0;
	else
		tmp = sqrt(Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 6.0 * ((y / x) / x);
	tmp = 0.0;
	if (y <= -660.0)
		tmp = t_0;
	elseif (y <= 330.0)
		tmp = y / (x / sin(x));
	elseif (y <= 1.42e+135)
		tmp = t_0;
	else
		tmp = sqrt((y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(6.0 * N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -660.0], t$95$0, If[LessEqual[y, 330.0], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.42e+135], t$95$0, N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -660 or 330 < y < 1.41999999999999998e135

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 4.5%

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

      1. associate-/l* 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in x around 0 3.8%

      \[\leadsto \frac{y}{\color{blue}{1 + 0.16666666666666666 \cdot {x}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 3.8%

         \[\leadsto \frac{y}{1 + \color{blue}{{x}^{2} \cdot 0.16666666666666666}} \]

      2. unpow2 3.8%

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

   9. Simplified 3.8%

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

   10. Taylor expanded in x around inf 41.5%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 41.5%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

       2. associate-/r* 41.5%

          \[\leadsto 6 \cdot \color{blue}{\frac{\frac{y}{x}}{x}} \]

   12. Simplified 41.5%

       \[\leadsto \color{blue}{6 \cdot \frac{\frac{y}{x}}{x}} \]

   if -660 < y < 330

   1. Initial program 84.3%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 83.8%

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

      1. associate-/l* 99.4%

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

   6. Simplified 99.4%

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

   if 1.41999999999999998e135 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 5.1%

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

   5. Taylor expanded in x around 0 26.3%

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

      1. *-commutative 26.3%

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

   7. Simplified 26.3%

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

      1. div-inv 26.3%

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

      2. associate-*l* 5.0%

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

      3. div-inv 5.0%

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

      4. *-inverses 5.0%

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

      5. *-commutative 5.0%

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

      6. *-un-lft-identity 5.0%

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

      7. add-sqr-sqrt 5.0%

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

      8. sqrt-unprod 61.9%

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

   9. Applied egg-rr 61.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -660:\\ \;\;\;\;6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 330:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+135}:\\ \;\;\;\;6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \]

Alternative 6: 56.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \frac{\frac{y}{x}}{x}\\ t_1 := \frac{x}{\frac{x}{y}}\\ \mathbf{if}\;y \leq -480:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.175:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{+255}:\\ \;\;\;\;y + -0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 6.0 (/ (/ y x) x))) (t_1 (/ x (/ x y))))
   (if (<= y -480.0)
     t_0
     (if (<= y 0.175)
       t_1
       (if (<= y 2.8e+194)
         t_0
         (if (<= y 1.52e+255)
           (+ y (* -0.16666666666666666 (* y (* x x))))
           t_1))))))

C

double code(double x, double y) {
	double t_0 = 6.0 * ((y / x) / x);
	double t_1 = x / (x / y);
	double tmp;
	if (y <= -480.0) {
		tmp = t_0;
	} else if (y <= 0.175) {
		tmp = t_1;
	} else if (y <= 2.8e+194) {
		tmp = t_0;
	} else if (y <= 1.52e+255) {
		tmp = y + (-0.16666666666666666 * (y * (x * x)));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 6.0d0 * ((y / x) / x)
    t_1 = x / (x / y)
    if (y <= (-480.0d0)) then
        tmp = t_0
    else if (y <= 0.175d0) then
        tmp = t_1
    else if (y <= 2.8d+194) then
        tmp = t_0
    else if (y <= 1.52d+255) then
        tmp = y + ((-0.16666666666666666d0) * (y * (x * x)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 6.0 * ((y / x) / x);
	double t_1 = x / (x / y);
	double tmp;
	if (y <= -480.0) {
		tmp = t_0;
	} else if (y <= 0.175) {
		tmp = t_1;
	} else if (y <= 2.8e+194) {
		tmp = t_0;
	} else if (y <= 1.52e+255) {
		tmp = y + (-0.16666666666666666 * (y * (x * x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 6.0 * ((y / x) / x)
	t_1 = x / (x / y)
	tmp = 0
	if y <= -480.0:
		tmp = t_0
	elif y <= 0.175:
		tmp = t_1
	elif y <= 2.8e+194:
		tmp = t_0
	elif y <= 1.52e+255:
		tmp = y + (-0.16666666666666666 * (y * (x * x)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(6.0 * Float64(Float64(y / x) / x))
	t_1 = Float64(x / Float64(x / y))
	tmp = 0.0
	if (y <= -480.0)
		tmp = t_0;
	elseif (y <= 0.175)
		tmp = t_1;
	elseif (y <= 2.8e+194)
		tmp = t_0;
	elseif (y <= 1.52e+255)
		tmp = Float64(y + Float64(-0.16666666666666666 * Float64(y * Float64(x * x))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 6.0 * ((y / x) / x);
	t_1 = x / (x / y);
	tmp = 0.0;
	if (y <= -480.0)
		tmp = t_0;
	elseif (y <= 0.175)
		tmp = t_1;
	elseif (y <= 2.8e+194)
		tmp = t_0;
	elseif (y <= 1.52e+255)
		tmp = y + (-0.16666666666666666 * (y * (x * x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(6.0 * N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -480.0], t$95$0, If[LessEqual[y, 0.175], t$95$1, If[LessEqual[y, 2.8e+194], t$95$0, If[LessEqual[y, 1.52e+255], N[(y + N[(-0.16666666666666666 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -480 or 0.17499999999999999 < y < 2.8000000000000001e194

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 4.6%

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

      1. associate-/l* 4.6%

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

   6. Simplified 4.6%

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

   7. Taylor expanded in x around 0 3.8%

      \[\leadsto \frac{y}{\color{blue}{1 + 0.16666666666666666 \cdot {x}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 3.8%

         \[\leadsto \frac{y}{1 + \color{blue}{{x}^{2} \cdot 0.16666666666666666}} \]

      2. unpow2 3.8%

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

   9. Simplified 3.8%

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

   10. Taylor expanded in x around inf 40.1%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 40.1%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

       2. associate-/r* 40.2%

          \[\leadsto 6 \cdot \color{blue}{\frac{\frac{y}{x}}{x}} \]

   12. Simplified 40.2%

       \[\leadsto \color{blue}{6 \cdot \frac{\frac{y}{x}}{x}} \]

   if -480 < y < 0.17499999999999999 or 1.52000000000000001e255 < y

   1. Initial program 85.1%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 79.6%

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

   5. Taylor expanded in x around 0 39.2%

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

      1. *-commutative 39.2%

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

   7. Simplified 39.2%

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

      1. div-inv 39.1%

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

      2. associate-*l* 51.7%

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

      3. div-inv 51.9%

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

      4. *-inverses 51.9%

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

      5. *-commutative 51.9%

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

      6. *-un-lft-identity 51.9%

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

      7. log1p-expm1-u 57.3%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]

   9. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 51.9%

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

       2. *-un-lft-identity 51.9%

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

       3. *-commutative 51.9%

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

       4. metadata-eval 51.9%

          \[\leadsto y \cdot \color{blue}{{x}^{0}} \]

       5. metadata-eval 51.9%

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

       6. pow-prod-up 51.7%

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

       7. inv-pow 51.7%

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

       8. pow1 51.7%

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

       9. associate-*l* 78.9%

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

       10. div-inv 79.0%

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

       11. *-commutative 79.0%

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

       12. associate-*r/ 39.2%

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

       13. associate-/l* 80.6%

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

   11. Applied egg-rr 80.6%

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

   if 2.8000000000000001e194 < y < 1.52000000000000001e255

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 3.7%

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

      1. associate-/l* 3.7%

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

      2. associate-/r/ 13.1%

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

   6. Simplified 13.1%

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

   7. Taylor expanded in x around 0 70.5%

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

      1. *-commutative 70.5%

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

      2. unpow2 70.5%

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

   9. Simplified 70.5%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -480:\\ \;\;\;\;6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 0.175:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+194}:\\ \;\;\;\;6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{+255}:\\ \;\;\;\;y + -0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \]

Alternative 7: 57.0% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{if}\;y \leq -260:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.175:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+195}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+248}:\\ \;\;\;\;y + -0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 6.0 (/ (/ y x) x))))
   (if (<= y -260.0)
     t_0
     (if (<= y 0.175)
       (/ y (+ 1.0 (* x (* x 0.16666666666666666))))
       (if (<= y 7.2e+195)
         t_0
         (if (<= y 5.6e+248)
           (+ y (* -0.16666666666666666 (* y (* x x))))
           (/ x (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = 6.0 * ((y / x) / x);
	double tmp;
	if (y <= -260.0) {
		tmp = t_0;
	} else if (y <= 0.175) {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	} else if (y <= 7.2e+195) {
		tmp = t_0;
	} else if (y <= 5.6e+248) {
		tmp = y + (-0.16666666666666666 * (y * (x * x)));
	} else {
		tmp = x / (x / y);
	}
	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 = 6.0d0 * ((y / x) / x)
    if (y <= (-260.0d0)) then
        tmp = t_0
    else if (y <= 0.175d0) then
        tmp = y / (1.0d0 + (x * (x * 0.16666666666666666d0)))
    else if (y <= 7.2d+195) then
        tmp = t_0
    else if (y <= 5.6d+248) then
        tmp = y + ((-0.16666666666666666d0) * (y * (x * x)))
    else
        tmp = x / (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 6.0 * ((y / x) / x);
	double tmp;
	if (y <= -260.0) {
		tmp = t_0;
	} else if (y <= 0.175) {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	} else if (y <= 7.2e+195) {
		tmp = t_0;
	} else if (y <= 5.6e+248) {
		tmp = y + (-0.16666666666666666 * (y * (x * x)));
	} else {
		tmp = x / (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 6.0 * ((y / x) / x)
	tmp = 0
	if y <= -260.0:
		tmp = t_0
	elif y <= 0.175:
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)))
	elif y <= 7.2e+195:
		tmp = t_0
	elif y <= 5.6e+248:
		tmp = y + (-0.16666666666666666 * (y * (x * x)))
	else:
		tmp = x / (x / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(6.0 * Float64(Float64(y / x) / x))
	tmp = 0.0
	if (y <= -260.0)
		tmp = t_0;
	elseif (y <= 0.175)
		tmp = Float64(y / Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666))));
	elseif (y <= 7.2e+195)
		tmp = t_0;
	elseif (y <= 5.6e+248)
		tmp = Float64(y + Float64(-0.16666666666666666 * Float64(y * Float64(x * x))));
	else
		tmp = Float64(x / Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 6.0 * ((y / x) / x);
	tmp = 0.0;
	if (y <= -260.0)
		tmp = t_0;
	elseif (y <= 0.175)
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	elseif (y <= 7.2e+195)
		tmp = t_0;
	elseif (y <= 5.6e+248)
		tmp = y + (-0.16666666666666666 * (y * (x * x)));
	else
		tmp = x / (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(6.0 * N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -260.0], t$95$0, If[LessEqual[y, 0.175], N[(y / N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e+195], t$95$0, If[LessEqual[y, 5.6e+248], N[(y + N[(-0.16666666666666666 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -260 or 0.17499999999999999 < y < 7.1999999999999997e195

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 4.6%

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

      1. associate-/l* 4.6%

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

   6. Simplified 4.6%

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

   7. Taylor expanded in x around 0 3.8%

      \[\leadsto \frac{y}{\color{blue}{1 + 0.16666666666666666 \cdot {x}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 3.8%

         \[\leadsto \frac{y}{1 + \color{blue}{{x}^{2} \cdot 0.16666666666666666}} \]

      2. unpow2 3.8%

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

   9. Simplified 3.8%

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

   10. Taylor expanded in x around inf 40.1%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 40.1%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

       2. associate-/r* 40.2%

          \[\leadsto 6 \cdot \color{blue}{\frac{\frac{y}{x}}{x}} \]

   12. Simplified 40.2%

       \[\leadsto \color{blue}{6 \cdot \frac{\frac{y}{x}}{x}} \]

   if -260 < y < 0.17499999999999999

   1. Initial program 84.2%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 84.2%

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

      1. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 82.3%

      \[\leadsto \frac{y}{\color{blue}{1 + 0.16666666666666666 \cdot {x}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 82.3%

         \[\leadsto \frac{y}{1 + \color{blue}{{x}^{2} \cdot 0.16666666666666666}} \]

      2. unpow2 82.3%

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

   9. Simplified 82.3%

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

   10. Taylor expanded in x around 0 82.3%

       \[\leadsto \frac{y}{1 + \color{blue}{0.16666666666666666 \cdot {x}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 82.3%

          \[\leadsto \frac{y}{1 + \color{blue}{{x}^{2} \cdot 0.16666666666666666}} \]

       2. unpow2 82.3%

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

       3. associate-*r* 82.3%

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

   12. Simplified 82.3%

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

   if 7.1999999999999997e195 < y < 5.6000000000000004e248

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 3.7%

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

      1. associate-/l* 3.7%

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

      2. associate-/r/ 13.1%

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

   6. Simplified 13.1%

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

   7. Taylor expanded in x around 0 70.5%

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

      1. *-commutative 70.5%

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

      2. unpow2 70.5%

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

   9. Simplified 70.5%

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

   if 5.6000000000000004e248 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 8.0%

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

   5. Taylor expanded in x around 0 43.7%

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

      1. *-commutative 43.7%

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

   7. Simplified 43.7%

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

      1. div-inv 43.7%

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

      2. associate-*l* 9.5%

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

      3. div-inv 9.5%

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

      4. *-inverses 9.5%

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

      5. *-commutative 9.5%

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

      6. *-un-lft-identity 9.5%

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

      7. log1p-expm1-u 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 9.5%

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

       2. *-un-lft-identity 9.5%

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

       3. *-commutative 9.5%

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

       4. metadata-eval 9.5%

          \[\leadsto y \cdot \color{blue}{{x}^{0}} \]

       5. metadata-eval 9.5%

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

       6. pow-prod-up 9.5%

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

       7. inv-pow 9.5%

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

       8. pow1 9.5%

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

       9. associate-*l* 65.8%

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

       10. div-inv 65.8%

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

       11. *-commutative 65.8%

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

       12. associate-*r/ 43.7%

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

       13. associate-/l* 65.8%

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

   11. Applied egg-rr 65.8%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -260:\\ \;\;\;\;6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 0.175:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+195}:\\ \;\;\;\;6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+248}:\\ \;\;\;\;y + -0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \]

Alternative 8: 56.7% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \frac{\frac{y}{x}}{x}\\ t_1 := \frac{x}{\frac{x}{y}}\\ \mathbf{if}\;y \leq -310:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.175:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+225}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+248}:\\ \;\;\;\;y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 6.0 (/ (/ y x) x))) (t_1 (/ x (/ x y))))
   (if (<= y -310.0)
     t_0
     (if (<= y 0.175)
       t_1
       (if (<= y 6e+225)
         t_0
         (if (<= y 5.2e+248) (* y (* x (* x -0.16666666666666666))) t_1))))))

C

double code(double x, double y) {
	double t_0 = 6.0 * ((y / x) / x);
	double t_1 = x / (x / y);
	double tmp;
	if (y <= -310.0) {
		tmp = t_0;
	} else if (y <= 0.175) {
		tmp = t_1;
	} else if (y <= 6e+225) {
		tmp = t_0;
	} else if (y <= 5.2e+248) {
		tmp = y * (x * (x * -0.16666666666666666));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 6.0d0 * ((y / x) / x)
    t_1 = x / (x / y)
    if (y <= (-310.0d0)) then
        tmp = t_0
    else if (y <= 0.175d0) then
        tmp = t_1
    else if (y <= 6d+225) then
        tmp = t_0
    else if (y <= 5.2d+248) then
        tmp = y * (x * (x * (-0.16666666666666666d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 6.0 * ((y / x) / x);
	double t_1 = x / (x / y);
	double tmp;
	if (y <= -310.0) {
		tmp = t_0;
	} else if (y <= 0.175) {
		tmp = t_1;
	} else if (y <= 6e+225) {
		tmp = t_0;
	} else if (y <= 5.2e+248) {
		tmp = y * (x * (x * -0.16666666666666666));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 6.0 * ((y / x) / x)
	t_1 = x / (x / y)
	tmp = 0
	if y <= -310.0:
		tmp = t_0
	elif y <= 0.175:
		tmp = t_1
	elif y <= 6e+225:
		tmp = t_0
	elif y <= 5.2e+248:
		tmp = y * (x * (x * -0.16666666666666666))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(6.0 * Float64(Float64(y / x) / x))
	t_1 = Float64(x / Float64(x / y))
	tmp = 0.0
	if (y <= -310.0)
		tmp = t_0;
	elseif (y <= 0.175)
		tmp = t_1;
	elseif (y <= 6e+225)
		tmp = t_0;
	elseif (y <= 5.2e+248)
		tmp = Float64(y * Float64(x * Float64(x * -0.16666666666666666)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 6.0 * ((y / x) / x);
	t_1 = x / (x / y);
	tmp = 0.0;
	if (y <= -310.0)
		tmp = t_0;
	elseif (y <= 0.175)
		tmp = t_1;
	elseif (y <= 6e+225)
		tmp = t_0;
	elseif (y <= 5.2e+248)
		tmp = y * (x * (x * -0.16666666666666666));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(6.0 * N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -310.0], t$95$0, If[LessEqual[y, 0.175], t$95$1, If[LessEqual[y, 6e+225], t$95$0, If[LessEqual[y, 5.2e+248], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -310 or 0.17499999999999999 < y < 6.000000000000001e225

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 4.6%

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

      1. associate-/l* 4.6%

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

   6. Simplified 4.6%

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

   7. Taylor expanded in x around 0 3.8%

      \[\leadsto \frac{y}{\color{blue}{1 + 0.16666666666666666 \cdot {x}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 3.8%

         \[\leadsto \frac{y}{1 + \color{blue}{{x}^{2} \cdot 0.16666666666666666}} \]

      2. unpow2 3.8%

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

   9. Simplified 3.8%

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

   10. Taylor expanded in x around inf 40.0%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 40.0%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

       2. associate-/r* 40.0%

          \[\leadsto 6 \cdot \color{blue}{\frac{\frac{y}{x}}{x}} \]

   12. Simplified 40.0%

       \[\leadsto \color{blue}{6 \cdot \frac{\frac{y}{x}}{x}} \]

   if -310 < y < 0.17499999999999999 or 5.20000000000000019e248 < y

   1. Initial program 85.1%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 79.6%

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

   5. Taylor expanded in x around 0 39.2%

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

      1. *-commutative 39.2%

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

   7. Simplified 39.2%

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

      1. div-inv 39.1%

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

      2. associate-*l* 51.7%

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

      3. div-inv 51.9%

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

      4. *-inverses 51.9%

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

      5. *-commutative 51.9%

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

      6. *-un-lft-identity 51.9%

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

      7. log1p-expm1-u 57.3%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]

   9. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 51.9%

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

       2. *-un-lft-identity 51.9%

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

       3. *-commutative 51.9%

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

       4. metadata-eval 51.9%

          \[\leadsto y \cdot \color{blue}{{x}^{0}} \]

       5. metadata-eval 51.9%

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

       6. pow-prod-up 51.7%

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

       7. inv-pow 51.7%

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

       8. pow1 51.7%

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

       9. associate-*l* 78.9%

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

       10. div-inv 79.0%

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

       11. *-commutative 79.0%

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

       12. associate-*r/ 39.2%

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

       13. associate-/l* 80.6%

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

   11. Applied egg-rr 80.6%

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

   if 6.000000000000001e225 < y < 5.20000000000000019e248

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 3.4%

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

      1. associate-/l* 3.4%

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

      2. associate-/r/ 3.4%

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

   6. Simplified 3.4%

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

   7. Taylor expanded in x around 0 85.7%

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

      1. +-commutative 85.7%

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

      2. remove-double-neg 85.7%

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

      3. unsub-neg 85.7%

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

      4. associate-*r* 85.7%

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

      5. *-commutative 85.7%

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

      6. fma-neg 85.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.16666666666666666 \cdot {x}^{2}, -\left(-y\right)\right)} \]

      7. unpow2 85.7%

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

      8. associate-*r* 85.7%

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

      9. remove-double-neg 85.7%

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

   9. Simplified 85.7%

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

   10. Taylor expanded in x around inf 85.7%

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

       1. associate-*r* 85.7%

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

       2. *-commutative 85.7%

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

       3. unpow2 85.7%

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

       4. associate-*r* 85.7%

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

   12. Simplified 85.7%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -310:\\ \;\;\;\;6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 0.175:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+225}:\\ \;\;\;\;6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+248}:\\ \;\;\;\;y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \]

Alternative 9: 57.4% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -440 \lor \neg \left(y \leq 0.175\right):\\ \;\;\;\;6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -440.0) (not (<= y 0.175)))
   (* 6.0 (/ (/ y x) x))
   (/ x (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -440.0) || !(y <= 0.175)) {
		tmp = 6.0 * ((y / x) / x);
	} else {
		tmp = x / (x / 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 <= (-440.0d0)) .or. (.not. (y <= 0.175d0))) then
        tmp = 6.0d0 * ((y / x) / x)
    else
        tmp = x / (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -440.0) || !(y <= 0.175)) {
		tmp = 6.0 * ((y / x) / x);
	} else {
		tmp = x / (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -440.0) or not (y <= 0.175):
		tmp = 6.0 * ((y / x) / x)
	else:
		tmp = x / (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -440.0) || !(y <= 0.175))
		tmp = Float64(6.0 * Float64(Float64(y / x) / x));
	else
		tmp = Float64(x / Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -440.0) || ~((y <= 0.175)))
		tmp = 6.0 * ((y / x) / x);
	else
		tmp = x / (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -440.0], N[Not[LessEqual[y, 0.175]], $MachinePrecision]], N[(6.0 * N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -440 or 0.17499999999999999 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 4.8%

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

      1. associate-/l* 4.8%

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

   6. Simplified 4.8%

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

   7. Taylor expanded in x around 0 3.9%

      \[\leadsto \frac{y}{\color{blue}{1 + 0.16666666666666666 \cdot {x}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 3.9%

         \[\leadsto \frac{y}{1 + \color{blue}{{x}^{2} \cdot 0.16666666666666666}} \]

      2. unpow2 3.9%

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

   9. Simplified 3.9%

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

   10. Taylor expanded in x around inf 39.4%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 39.4%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

       2. associate-/r* 39.4%

          \[\leadsto 6 \cdot \color{blue}{\frac{\frac{y}{x}}{x}} \]

   12. Simplified 39.4%

       \[\leadsto \color{blue}{6 \cdot \frac{\frac{y}{x}}{x}} \]

   if -440 < y < 0.17499999999999999

   1. Initial program 84.2%

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

      1. associate-*r/ 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. Taylor expanded in y around 0 84.2%

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

   5. Taylor expanded in x around 0 38.9%

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

      1. *-commutative 38.9%

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

   7. Simplified 38.9%

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

      1. div-inv 38.8%

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

      2. associate-*l* 54.4%

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

      3. div-inv 54.6%

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

      4. *-inverses 54.6%

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

      5. *-commutative 54.6%

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

      6. *-un-lft-identity 54.6%

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

      7. log1p-expm1-u 54.6%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]

   9. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 54.6%

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

       2. *-un-lft-identity 54.6%

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

       3. *-commutative 54.6%

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

       4. metadata-eval 54.6%

          \[\leadsto y \cdot \color{blue}{{x}^{0}} \]

       5. metadata-eval 54.6%

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

       6. pow-prod-up 54.4%

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

       7. inv-pow 54.4%

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

       8. pow1 54.4%

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

       9. associate-*l* 79.8%

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

       10. div-inv 79.9%

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

       11. *-commutative 79.9%

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

       12. associate-*r/ 38.9%

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

       13. associate-/l* 81.6%

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

   11. Applied egg-rr 81.6%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -440 \lor \neg \left(y \leq 0.175\right):\\ \;\;\;\;6 \cdot \frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \]

Alternative 10: 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 92.3%

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

   1. associate-*r/ 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. Taylor expanded in y around 0 43.6%

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

5. Taylor expanded in x around 0 27.4%

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

   1. *-commutative 27.4%

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

7. Simplified 27.4%

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

   1. associate-/l* 28.9%

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

   2. associate-/r/ 49.5%

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

9. Applied egg-rr 49.5%

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

10. Final simplification 49.5%

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

Alternative 11: 50.0% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return x / (x / y);
}

Python

def code(x, y):
	return x / (x / y)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

   1. associate-*r/ 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. Taylor expanded in y around 0 43.6%

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

5. Taylor expanded in x around 0 27.4%

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

   1. *-commutative 27.4%

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

7. Simplified 27.4%

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

   1. div-inv 27.4%

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

   2. associate-*l* 28.8%

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

   3. div-inv 28.9%

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

   4. *-inverses 28.9%

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

   5. *-commutative 28.9%

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

   6. *-un-lft-identity 28.9%

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

   7. log1p-expm1-u 66.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]

9. Applied egg-rr 66.1%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]
10. Step-by-step derivation

    1. log1p-expm1-u 28.9%

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

    2. *-un-lft-identity 28.9%

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

    3. *-commutative 28.9%

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

    4. metadata-eval 28.9%

       \[\leadsto y \cdot \color{blue}{{x}^{0}} \]

    5. metadata-eval 28.9%

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

    6. pow-prod-up 28.8%

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

    7. inv-pow 28.8%

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

    8. pow1 28.8%

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

    9. associate-*l* 49.4%

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

    10. div-inv 49.5%

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

    11. *-commutative 49.5%

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

    12. associate-*r/ 27.4%

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

    13. associate-/l* 49.9%

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

11. Applied egg-rr 49.9%

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

12. Final simplification 49.9%

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

Alternative 12: 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 92.3%

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

   1. associate-*r/ 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. Taylor expanded in y around 0 43.6%

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

   1. associate-/l* 51.2%

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

   2. associate-/r/ 59.4%

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

6. Simplified 59.4%

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

7. Taylor expanded in x around 0 28.9%

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

8. Final simplification 28.9%

   \[\leadsto y \]

Developer target: 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 2023271 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (* (sin x) (/ (sinh y) x))

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