Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.0% → 99.8%

Time: 9.2s

Alternatives: 18

Speedup: 1.0×

• 50.1% 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: 89.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 86.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(1 + y\right) - e^{-y}\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 10^{-8}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(e^{y} - \left(1 - y\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -4e+20)
     (* (- (+ 1.0 y) (exp (- y))) 0.5)
     (if (<= t_0 1e-8) (* (/ (sin x) x) y) (* (- (exp y) (- 1.0 y)) 0.5)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = ((1.0 + y) - exp(-y)) * 0.5;
	} else if (t_0 <= 1e-8) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = (exp(y) - (1.0 - y)) * 0.5;
	}
	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 = (sin(x) * sinh(y)) / x
    if (t_0 <= (-4d+20)) then
        tmp = ((1.0d0 + y) - exp(-y)) * 0.5d0
    else if (t_0 <= 1d-8) then
        tmp = (sin(x) / x) * y
    else
        tmp = (exp(y) - (1.0d0 - y)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (Math.sin(x) * Math.sinh(y)) / x;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = ((1.0 + y) - Math.exp(-y)) * 0.5;
	} else if (t_0 <= 1e-8) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = (Math.exp(y) - (1.0 - y)) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sin(x) * math.sinh(y)) / x
	tmp = 0
	if t_0 <= -4e+20:
		tmp = ((1.0 + y) - math.exp(-y)) * 0.5
	elif t_0 <= 1e-8:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = (math.exp(y) - (1.0 - y)) * 0.5
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -4e+20)
		tmp = Float64(Float64(Float64(1.0 + y) - exp(Float64(-y))) * 0.5);
	elseif (t_0 <= 1e-8)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(Float64(exp(y) - Float64(1.0 - y)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sin(x) * sinh(y)) / x;
	tmp = 0.0;
	if (t_0 <= -4e+20)
		tmp = ((1.0 + y) - exp(-y)) * 0.5;
	elseif (t_0 <= 1e-8)
		tmp = (sin(x) / x) * y;
	else
		tmp = (exp(y) - (1.0 - y)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+20], N[(N[(N[(1.0 + y), $MachinePrecision] - N[Exp[(-y)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 1e-8], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[Exp[y], $MachinePrecision] - N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4e20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 70.5

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 70.6%

      \[\leadsto \left(\left(1 + y\right) - e^{-y}\right) \cdot 0.5 \]

   if -4e20 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e-8

   1. Initial program 70.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 1e-8 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 77.0

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 77.1%

      \[\leadsto \left(e^{y} - \left(1 - y\right)\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5e-14)
   (sinh y)
   (/
    (*
     (*
      (sin x)
      (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0))
     y)
    x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e-14) {
		tmp = sinh(y);
	} else {
		tmp = ((sin(x) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0)) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e-14)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(Float64(sin(x) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e-14], N[Sinh[y], $MachinePrecision], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e-14

   1. Initial program 81.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 52.3

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.1%

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

   if 5.0000000000000002e-14 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right) \cdot y}}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right) \cdot y}}{x} \]

   5. Applied rewrites 82.7%

      \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)}{x} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5e-14)
   (sinh y)
   (*
    (/
     (*
      (sin x)
      (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0))
     x)
    y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e-14) {
		tmp = sinh(y);
	} else {
		tmp = ((sin(x) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0)) / x) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e-14)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(Float64(sin(x) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)) / x) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e-14], N[Sinh[y], $MachinePrecision], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e-14

   1. Initial program 81.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 52.3

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.1%

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

   if 5.0000000000000002e-14 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

   5. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)}{x} \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 77.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x} \cdot \left(\left(\sin x \cdot \mathsf{fma}\left(0.08333333333333333, y \cdot y, 0.5\right)\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5e-14)
   (sinh y)
   (* (/ 2.0 x) (* (* (sin x) (fma 0.08333333333333333 (* y y) 0.5)) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e-14) {
		tmp = sinh(y);
	} else {
		tmp = (2.0 / x) * ((sin(x) * fma(0.08333333333333333, (y * y), 0.5)) * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e-14)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(2.0 / x) * Float64(Float64(sin(x) * fma(0.08333333333333333, Float64(y * y), 0.5)) * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e-14], N[Sinh[y], $MachinePrecision], N[(N[(2.0 / x), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[(0.08333333333333333 * N[(y * y), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e-14

   1. Initial program 81.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 52.3

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.1%

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

   if 5.0000000000000002e-14 < x

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-sinh.f64 N/A

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

      5. sinh-def N/A

         \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}} \]

      6. frac-times N/A

         \[\leadsto \color{blue}{\frac{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{x \cdot 2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \sin x}}{x \cdot 2} \]

      8. sinh-undef N/A

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

      9. lift-sinh.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

          \[\leadsto \color{blue}{\frac{2}{x} \cdot \frac{\sin x \cdot \sinh y}{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{x} \cdot \frac{\sin x \cdot \sinh y}{2}} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{x}} \cdot \frac{\sin x \cdot \sinh y}{2} \]

      16. lower-/.f64 99.9

          \[\leadsto \frac{2}{x} \cdot \color{blue}{\frac{\sin x \cdot \sinh y}{2}} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{2}{x} \cdot \frac{\color{blue}{\sin x \cdot \sinh y}}{2} \]

      18. *-commutative N/A

          \[\leadsto \frac{2}{x} \cdot \frac{\color{blue}{\sinh y \cdot \sin x}}{2} \]

      19. lower-*.f64 99.9

          \[\leadsto \frac{2}{x} \cdot \frac{\color{blue}{\sinh y \cdot \sin x}}{2} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{2}{x} \cdot \frac{\sinh y \cdot \sin x}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2}{x} \cdot \color{blue}{\left(y \cdot \left(\frac{1}{12} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{2} \cdot \sin x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{x} \cdot \color{blue}{\left(\left(\frac{1}{12} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{2} \cdot \sin x\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{x} \cdot \left(\left(\frac{1}{12} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)} + \frac{1}{2} \cdot \sin x\right) \cdot y\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{x} \cdot \left(\left(\color{blue}{\left(\frac{1}{12} \cdot \sin x\right) \cdot {y}^{2}} + \frac{1}{2} \cdot \sin x\right) \cdot y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{x} \cdot \color{blue}{\left(\left(\left(\frac{1}{12} \cdot \sin x\right) \cdot {y}^{2} + \frac{1}{2} \cdot \sin x\right) \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{x} \cdot \left(\left(\color{blue}{\frac{1}{12} \cdot \left(\sin x \cdot {y}^{2}\right)} + \frac{1}{2} \cdot \sin x\right) \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{x} \cdot \left(\left(\frac{1}{12} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)} + \frac{1}{2} \cdot \sin x\right) \cdot y\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{x} \cdot \left(\left(\color{blue}{\left(\frac{1}{12} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{2} \cdot \sin x\right) \cdot y\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \frac{2}{x} \cdot \left(\color{blue}{\left(\sin x \cdot \left(\frac{1}{12} \cdot {y}^{2} + \frac{1}{2}\right)\right)} \cdot y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{x} \cdot \left(\color{blue}{\left(\sin x \cdot \left(\frac{1}{12} \cdot {y}^{2} + \frac{1}{2}\right)\right)} \cdot y\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \frac{2}{x} \cdot \left(\left(\color{blue}{\sin x} \cdot \left(\frac{1}{12} \cdot {y}^{2} + \frac{1}{2}\right)\right) \cdot y\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{2}{x} \cdot \left(\left(\sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{12}, {y}^{2}, \frac{1}{2}\right)}\right) \cdot y\right) \]

      12. unpow2 N/A

          \[\leadsto \frac{2}{x} \cdot \left(\left(\sin x \cdot \mathsf{fma}\left(\frac{1}{12}, \color{blue}{y \cdot y}, \frac{1}{2}\right)\right) \cdot y\right) \]

      13. lower-*.f64 77.9

          \[\leadsto \frac{2}{x} \cdot \left(\left(\sin x \cdot \mathsf{fma}\left(0.08333333333333333, \color{blue}{y \cdot y}, 0.5\right)\right) \cdot y\right) \]

   7. Applied rewrites 77.9%

      \[\leadsto \frac{2}{x} \cdot \color{blue}{\left(\left(\sin x \cdot \mathsf{fma}\left(0.08333333333333333, y \cdot y, 0.5\right)\right) \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\right) \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5e-14)
   (sinh y)
   (* (* (fma (* y y) 0.16666666666666666 1.0) (sin x)) (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e-14) {
		tmp = sinh(y);
	} else {
		tmp = (fma((y * y), 0.16666666666666666, 1.0) * sin(x)) * (y / x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e-14)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * sin(x)) * Float64(y / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e-14], N[Sinh[y], $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e-14

   1. Initial program 81.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 52.3

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.1%

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

   if 5.0000000000000002e-14 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

   5. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)}{x} \cdot y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right)}{x} \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 76.4%

      \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)}{x} \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 76.4%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\right) \cdot \color{blue}{\frac{y}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 67.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{+82}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot \mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right)\right) \cdot x, x, -0.16666666666666666 \cdot y\right), x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left({y}^{4} \cdot 0.008333333333333333\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.55e+82)
   (sinh y)
   (if (<= x 1.45e+146)
     (fma
      (fma
       (* (* y (fma -0.0001984126984126984 (* x x) 0.008333333333333333)) x)
       x
       (* -0.16666666666666666 y))
      (* x x)
      y)
     (* (* (pow y 4.0) 0.008333333333333333) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.55e+82) {
		tmp = sinh(y);
	} else if (x <= 1.45e+146) {
		tmp = fma(fma(((y * fma(-0.0001984126984126984, (x * x), 0.008333333333333333)) * x), x, (-0.16666666666666666 * y)), (x * x), y);
	} else {
		tmp = (pow(y, 4.0) * 0.008333333333333333) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.55e+82)
		tmp = sinh(y);
	elseif (x <= 1.45e+146)
		tmp = fma(fma(Float64(Float64(y * fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333)) * x), x, Float64(-0.16666666666666666 * y)), Float64(x * x), y);
	else
		tmp = Float64(Float64((y ^ 4.0) * 0.008333333333333333) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.55e+82], N[Sinh[y], $MachinePrecision], If[LessEqual[x, 1.45e+146], N[(N[(N[(N[(y * N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x + N[(-0.16666666666666666 * y), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], N[(N[(N[Power[y, 4.0], $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.5500000000000001e82

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 51.8

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.1%

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

   if 2.5500000000000001e82 < x < 1.4499999999999999e146

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 38.3

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto y + \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot y + {x}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{120} \cdot y\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot \mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right)\right) \cdot x, x, -0.16666666666666666 \cdot y\right), \color{blue}{x \cdot x}, y\right) \]

   if 1.4499999999999999e146 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 23.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\frac{1}{120} \cdot {y}^{4}\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 54.0%

       \[\leadsto \left({y}^{4} \cdot 0.008333333333333333\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{+82}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot \mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right)\right) \cdot x, x, -0.16666666666666666 \cdot y\right), x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.55e+82)
   (sinh y)
   (if (<= x 1.15e+146)
     (fma
      (fma
       (* (* y (fma -0.0001984126984126984 (* x x) 0.008333333333333333)) x)
       x
       (* -0.16666666666666666 y))
      (* x x)
      y)
     (* (- (+ 1.0 y) (fma (- (* 0.5 y) 1.0) y 1.0)) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.55e+82) {
		tmp = sinh(y);
	} else if (x <= 1.15e+146) {
		tmp = fma(fma(((y * fma(-0.0001984126984126984, (x * x), 0.008333333333333333)) * x), x, (-0.16666666666666666 * y)), (x * x), y);
	} else {
		tmp = ((1.0 + y) - fma(((0.5 * y) - 1.0), y, 1.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.55e+82)
		tmp = sinh(y);
	elseif (x <= 1.15e+146)
		tmp = fma(fma(Float64(Float64(y * fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333)) * x), x, Float64(-0.16666666666666666 * y)), Float64(x * x), y);
	else
		tmp = Float64(Float64(Float64(1.0 + y) - fma(Float64(Float64(0.5 * y) - 1.0), y, 1.0)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.55e+82], N[Sinh[y], $MachinePrecision], If[LessEqual[x, 1.15e+146], N[(N[(N[(N[(y * N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x + N[(-0.16666666666666666 * y), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(N[(N[(0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.5500000000000001e82

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 51.8

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.1%

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

   if 2.5500000000000001e82 < x < 1.15e146

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 38.3

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto y + \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot y + {x}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{120} \cdot y\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot \mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right)\right) \cdot x, x, -0.16666666666666666 \cdot y\right), \color{blue}{x \cdot x}, y\right) \]

   if 1.15e146 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 67.4

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 52.8%

      \[\leadsto \left(e^{y} - \left(1 - y\right)\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.3%

       \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5 \]

   12. Taylor expanded in y around 0

       \[\leadsto \left(\left(1 + y\right) - \left(1 + y \cdot \left(\frac{1}{2} \cdot y - 1\right)\right)\right) \cdot \frac{1}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 55.1%

       \[\leadsto \left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 61.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+69}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot \mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right)\right) \cdot x, x, -0.16666666666666666 \cdot y\right), x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.7e+69)
   (*
    (*
     (fma
      (fma
       (fma 0.0003968253968253968 (* y y) 0.016666666666666666)
       (* y y)
       0.3333333333333333)
      (* y y)
      2.0)
     y)
    0.5)
   (if (<= x 1.15e+146)
     (fma
      (fma
       (* (* y (fma -0.0001984126984126984 (* x x) 0.008333333333333333)) x)
       x
       (* -0.16666666666666666 y))
      (* x x)
      y)
     (* (- (+ 1.0 y) (fma (- (* 0.5 y) 1.0) y 1.0)) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.7e+69) {
		tmp = (fma(fma(fma(0.0003968253968253968, (y * y), 0.016666666666666666), (y * y), 0.3333333333333333), (y * y), 2.0) * y) * 0.5;
	} else if (x <= 1.15e+146) {
		tmp = fma(fma(((y * fma(-0.0001984126984126984, (x * x), 0.008333333333333333)) * x), x, (-0.16666666666666666 * y)), (x * x), y);
	} else {
		tmp = ((1.0 + y) - fma(((0.5 * y) - 1.0), y, 1.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.7e+69)
		tmp = Float64(Float64(fma(fma(fma(0.0003968253968253968, Float64(y * y), 0.016666666666666666), Float64(y * y), 0.3333333333333333), Float64(y * y), 2.0) * y) * 0.5);
	elseif (x <= 1.15e+146)
		tmp = fma(fma(Float64(Float64(y * fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333)) * x), x, Float64(-0.16666666666666666 * y)), Float64(x * x), y);
	else
		tmp = Float64(Float64(Float64(1.0 + y) - fma(Float64(Float64(0.5 * y) - 1.0), y, 1.0)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.7e+69], N[(N[(N[(N[(N[(0.0003968253968253968 * N[(y * y), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.15e+146], N[(N[(N[(N[(y * N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x + N[(-0.16666666666666666 * y), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(N[(N[(0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.6999999999999998e69

   1. Initial program 82.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 52.3

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 70.3%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5 \]

   if 2.6999999999999998e69 < x < 1.15e146

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 40.3

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

   5. Applied rewrites 40.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto y + \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot y + {x}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{120} \cdot y\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot \mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right)\right) \cdot x, x, -0.16666666666666666 \cdot y\right), \color{blue}{x \cdot x}, y\right) \]

   if 1.15e146 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 67.4

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 52.8%

      \[\leadsto \left(e^{y} - \left(1 - y\right)\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.3%

       \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5 \]

   12. Taylor expanded in y around 0

       \[\leadsto \left(\left(1 + y\right) - \left(1 + y \cdot \left(\frac{1}{2} \cdot y - 1\right)\right)\right) \cdot \frac{1}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 55.1%

       \[\leadsto \left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 60.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot \mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right)\right) \cdot x, x, -0.16666666666666666 \cdot y\right), x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.7e+69)
   (*
    (fma
     (* -0.16666666666666666 x)
     x
     (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0))
    y)
   (if (<= x 1.15e+146)
     (fma
      (fma
       (* (* y (fma -0.0001984126984126984 (* x x) 0.008333333333333333)) x)
       x
       (* -0.16666666666666666 y))
      (* x x)
      y)
     (* (- (+ 1.0 y) (fma (- (* 0.5 y) 1.0) y 1.0)) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.7e+69) {
		tmp = fma((-0.16666666666666666 * x), x, fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0)) * y;
	} else if (x <= 1.15e+146) {
		tmp = fma(fma(((y * fma(-0.0001984126984126984, (x * x), 0.008333333333333333)) * x), x, (-0.16666666666666666 * y)), (x * x), y);
	} else {
		tmp = ((1.0 + y) - fma(((0.5 * y) - 1.0), y, 1.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.7e+69)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)) * y);
	elseif (x <= 1.15e+146)
		tmp = fma(fma(Float64(Float64(y * fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333)) * x), x, Float64(-0.16666666666666666 * y)), Float64(x * x), y);
	else
		tmp = Float64(Float64(Float64(1.0 + y) - fma(Float64(Float64(0.5 * y) - 1.0), y, 1.0)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.7e+69], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 1.15e+146], N[(N[(N[(N[(y * N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x + N[(-0.16666666666666666 * y), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(N[(N[(0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.6999999999999998e69

   1. Initial program 82.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 35.1%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot -0.16666666666666666\right) \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{6} \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 68.0%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if 2.6999999999999998e69 < x < 1.15e146

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 40.3

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

   5. Applied rewrites 40.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto y + \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot y + {x}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{120} \cdot y\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot \mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right)\right) \cdot x, x, -0.16666666666666666 \cdot y\right), \color{blue}{x \cdot x}, y\right) \]

   if 1.15e146 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 67.4

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 52.8%

      \[\leadsto \left(e^{y} - \left(1 - y\right)\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.3%

       \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5 \]

   12. Taylor expanded in y around 0

       \[\leadsto \left(\left(1 + y\right) - \left(1 + y \cdot \left(\frac{1}{2} \cdot y - 1\right)\right)\right) \cdot \frac{1}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 55.1%

       \[\leadsto \left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 60.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right) \cdot x\right) \cdot x - 0.16666666666666666, x \cdot x, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.7e+69)
   (*
    (fma
     (* -0.16666666666666666 x)
     x
     (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0))
    y)
   (if (<= x 1.15e+146)
     (*
      (fma
       (-
        (* (* (fma -0.0001984126984126984 (* x x) 0.008333333333333333) x) x)
        0.16666666666666666)
       (* x x)
       1.0)
      y)
     (* (- (+ 1.0 y) (fma (- (* 0.5 y) 1.0) y 1.0)) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.7e+69) {
		tmp = fma((-0.16666666666666666 * x), x, fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0)) * y;
	} else if (x <= 1.15e+146) {
		tmp = fma((((fma(-0.0001984126984126984, (x * x), 0.008333333333333333) * x) * x) - 0.16666666666666666), (x * x), 1.0) * y;
	} else {
		tmp = ((1.0 + y) - fma(((0.5 * y) - 1.0), y, 1.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.7e+69)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)) * y);
	elseif (x <= 1.15e+146)
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333) * x) * x) - 0.16666666666666666), Float64(x * x), 1.0) * y);
	else
		tmp = Float64(Float64(Float64(1.0 + y) - fma(Float64(Float64(0.5 * y) - 1.0), y, 1.0)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.7e+69], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 1.15e+146], N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(N[(N[(0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.6999999999999998e69

   1. Initial program 82.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 35.1%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot -0.16666666666666666\right) \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{6} \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 68.0%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if 2.6999999999999998e69 < x < 1.15e146

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 40.3

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

   5. Applied rewrites 40.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 54.4%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right) \cdot x\right) \cdot x - 0.16666666666666666, x \cdot x, 1\right) \cdot y \]

   if 1.15e146 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 67.4

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 52.8%

      \[\leadsto \left(e^{y} - \left(1 - y\right)\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.3%

       \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5 \]

   12. Taylor expanded in y around 0

       \[\leadsto \left(\left(1 + y\right) - \left(1 + y \cdot \left(\frac{1}{2} \cdot y - 1\right)\right)\right) \cdot \frac{1}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 55.1%

       \[\leadsto \left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 60.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.35e+26)
   (*
    (fma
     (* -0.16666666666666666 x)
     x
     (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0))
    y)
   (* (- (+ 1.0 y) (fma (- (* 0.5 y) 1.0) y 1.0)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.35e+26) {
		tmp = fma((-0.16666666666666666 * x), x, fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0)) * y;
	} else {
		tmp = ((1.0 + y) - fma(((0.5 * y) - 1.0), y, 1.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.35e+26)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)) * y);
	else
		tmp = Float64(Float64(Float64(1.0 + y) - fma(Float64(Float64(0.5 * y) - 1.0), y, 1.0)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.35e+26], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(N[(N[(0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e26

   1. Initial program 82.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

   5. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 36.3%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot -0.16666666666666666\right) \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{6} \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 68.8%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if 1.35e26 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 56.9

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 42.6%

      \[\leadsto \left(e^{y} - \left(1 - y\right)\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.7%

       \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5 \]

   12. Taylor expanded in y around 0

       \[\leadsto \left(\left(1 + y\right) - \left(1 + y \cdot \left(\frac{1}{2} \cdot y - 1\right)\right)\right) \cdot \frac{1}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 47.5%

       \[\leadsto \left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 60.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3e+104)
   (*
    (fma (* (fma (* 0.008333333333333333 y) y 0.16666666666666666) y) y 1.0)
    y)
   (* (- (+ 1.0 y) (fma (- (* 0.5 y) 1.0) y 1.0)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3e+104) {
		tmp = fma((fma((0.008333333333333333 * y), y, 0.16666666666666666) * y), y, 1.0) * y;
	} else {
		tmp = ((1.0 + y) - fma(((0.5 * y) - 1.0), y, 1.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3e+104)
		tmp = Float64(fma(Float64(fma(Float64(0.008333333333333333 * y), y, 0.16666666666666666) * y), y, 1.0) * y);
	else
		tmp = Float64(Float64(Float64(1.0 + y) - fma(Float64(Float64(0.5 * y) - 1.0), y, 1.0)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3e+104], N[(N[(N[(N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(N[(N[(0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.99999999999999969e104

   1. Initial program 83.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

   5. Applied rewrites 88.2%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 65.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y \]
   11. Step-by-step derivation

   12. Applied rewrites 65.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y \]

   if 2.99999999999999969e104 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 60.4

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 60.4%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 48.3%

      \[\leadsto \left(e^{y} - \left(1 - y\right)\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 34.2%

       \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5 \]

   12. Taylor expanded in y around 0

       \[\leadsto \left(\left(1 + y\right) - \left(1 + y \cdot \left(\frac{1}{2} \cdot y - 1\right)\right)\right) \cdot \frac{1}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 50.5%

       \[\leadsto \left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 56.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3e+104)
   (* (fma (* y y) 0.16666666666666666 1.0) y)
   (* (- (+ 1.0 y) (fma (- (* 0.5 y) 1.0) y 1.0)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3e+104) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * y;
	} else {
		tmp = ((1.0 + y) - fma(((0.5 * y) - 1.0), y, 1.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3e+104)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y);
	else
		tmp = Float64(Float64(Float64(1.0 + y) - fma(Float64(Float64(0.5 * y) - 1.0), y, 1.0)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3e+104], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(N[(N[(0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.99999999999999969e104

   1. Initial program 83.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

   5. Applied rewrites 88.2%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 61.1%

       \[\leadsto \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y \]

   if 2.99999999999999969e104 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 60.4

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 60.4%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 48.3%

      \[\leadsto \left(e^{y} - \left(1 - y\right)\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 34.2%

       \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5 \]

   12. Taylor expanded in y around 0

       \[\leadsto \left(\left(1 + y\right) - \left(1 + y \cdot \left(\frac{1}{2} \cdot y - 1\right)\right)\right) \cdot \frac{1}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 50.5%

       \[\leadsto \left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 53.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+213}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.55e+82)
   (* (fma (* y y) 0.16666666666666666 1.0) y)
   (if (<= x 4e+213)
     (* (fma -0.16666666666666666 (* x x) 1.0) y)
     (* (- (+ 1.0 y) (- 1.0 y)) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.55e+82) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * y;
	} else if (x <= 4e+213) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * y;
	} else {
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.55e+82)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y);
	elseif (x <= 4e+213)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * y);
	else
		tmp = Float64(Float64(Float64(1.0 + y) - Float64(1.0 - y)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.55e+82], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 4e+213], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.5500000000000001e82

   1. Initial program 82.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

   5. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 66.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 61.9%

       \[\leadsto \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y \]

   if 2.5500000000000001e82 < x < 3.99999999999999994e213

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 37.5

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

   5. Applied rewrites 37.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 33.3%

      \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y \]

   if 3.99999999999999994e213 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 68.7

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.7%

      \[\leadsto \left(e^{y} - \left(1 - y\right)\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 48.7%

       \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 37.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+213}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4e+213)
   (* (fma -0.16666666666666666 (* x x) 1.0) y)
   (* (- (+ 1.0 y) (- 1.0 y)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4e+213) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * y;
	} else {
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4e+213)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * y);
	else
		tmp = Float64(Float64(Float64(1.0 + y) - Float64(1.0 - y)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4e+213], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999994e213

   1. Initial program 85.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 48.7

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 40.8%

      \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y \]

   if 3.99999999999999994e213 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 68.7

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.7%

      \[\leadsto \left(e^{y} - \left(1 - y\right)\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 48.7%

       \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 33.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{+23}:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.05e+23) (* 1.0 y) (* (- (+ 1.0 y) (- 1.0 y)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.05e+23) {
		tmp = 1.0 * y;
	} else {
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.05d+23) then
        tmp = 1.0d0 * y
    else
        tmp = ((1.0d0 + y) - (1.0d0 - y)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.05e+23) {
		tmp = 1.0 * y;
	} else {
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.05e+23:
		tmp = 1.0 * y
	else:
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.05e+23)
		tmp = Float64(1.0 * y);
	else
		tmp = Float64(Float64(Float64(1.0 + y) - Float64(1.0 - y)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.05e+23)
		tmp = 1.0 * y;
	else
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.05e+23], N[(1.0 * y), $MachinePrecision], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.04999999999999998e23

   1. Initial program 82.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 36.7%

      \[\leadsto 1 \cdot y \]

   if 2.04999999999999998e23 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 56.9

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 42.6%

      \[\leadsto \left(e^{y} - \left(1 - y\right)\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.7%

       \[\leadsto \left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 27.9% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 1.0 y))

C

double code(double x, double y) {
	return 1.0 * y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 * y
end function

Java

public static double code(double x, double y) {
	return 1.0 * y;
}

Python

def code(x, y):
	return 1.0 * y

Julia

function code(x, y)
	return Float64(1.0 * y)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 * y;
end

Wolfram

code[x_, y_] := N[(1.0 * y), $MachinePrecision]

Derivation

1. Initial program 86.2%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 49.5

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

5. Applied rewrites 49.5%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 \cdot y \]
7. Step-by-step derivation

8. Applied rewrites 29.2%

   \[\leadsto 1 \cdot y \]
9. 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 2024338 
(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))