Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.3% → 99.8%

Time: 9.2s

Alternatives: 19

Speedup: 1.4×

• 49.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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: 89.3% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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 87.6%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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: 88.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(1 + y\right) - e^{-y}\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left|0.008333333333333333 \cdot y\right|, y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (* (- (+ 1.0 y) (exp (- y))) 0.5)
     (if (<= t_0 5e-52)
       (* (/ (sin x) x) y)
       (*
        (fma
         (fma (fabs (* 0.008333333333333333 y)) y 0.16666666666666666)
         (* y y)
         1.0)
        y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((1.0 + y) - exp(-y)) * 0.5;
	} else if (t_0 <= 5e-52) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = fma(fma(fabs((0.008333333333333333 * y)), y, 0.16666666666666666), (y * y), 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(1.0 + y) - exp(Float64(-y))) * 0.5);
	elseif (t_0 <= 5e-52)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(fma(fma(abs(Float64(0.008333333333333333 * y)), y, 0.16666666666666666), Float64(y * y), 1.0) * y);
	end
	return 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, (-Infinity)], N[(N[(N[(1.0 + y), $MachinePrecision] - N[Exp[(-y)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 5e-52], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[Abs[N[(0.008333333333333333 * y), $MachinePrecision]], $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -inf.0

   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 72.9

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

   5. Applied rewrites 72.9%

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

   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 72.9%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5e-52

   1. Initial program 76.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 5e-52 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) 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 79.0%

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

   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 67.1%

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

   10. Applied rewrites 80.9%

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

Alternative 3: 89.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5800 \lor \neg \left(y \leq 0.08 \lor \neg \left(y \leq 4.2 \cdot 10^{+116}\right)\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5800.0) (not (or (<= y 0.08) (not (<= y 4.2e+116)))))
   (sinh y)
   (* (* (fma (* y y) 0.16666666666666666 1.0) (/ (sin x) x)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5800.0) || !((y <= 0.08) || !(y <= 4.2e+116))) {
		tmp = sinh(y);
	} else {
		tmp = (fma((y * y), 0.16666666666666666, 1.0) * (sin(x) / x)) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5800.0) || !((y <= 0.08) || !(y <= 4.2e+116)))
		tmp = sinh(y);
	else
		tmp = Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * Float64(sin(x) / x)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5800.0], N[Not[Or[LessEqual[y, 0.08], N[Not[LessEqual[y, 4.2e+116]], $MachinePrecision]]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5800 or 0.0800000000000000017 < y < 4.2000000000000002e116

   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 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 80.7%

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

   9. Applied rewrites 80.7%

      \[\leadsto \sinh y \]

   if -5800 < y < 0.0800000000000000017 or 4.2000000000000002e116 < y

   1. Initial program 82.2%

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

   4. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \sinh \left(3 \cdot y\right)\right) \cdot \sin x}{\left(2 \cdot x\right) \cdot \mathsf{fma}\left(2, \cosh \left(2 \cdot y\right), 1\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   7. Applied rewrites 97.7%

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

   9. Applied rewrites 97.7%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5800 \lor \neg \left(y \leq 0.08 \lor \neg \left(y \leq 4.2 \cdot 10^{+116}\right)\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{if}\;y \leq -5800:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 0.08:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sin x}{x}\right) \cdot y\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+116}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \sin x}{x} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) 0.16666666666666666 1.0)))
   (if (<= y -5800.0)
     (sinh y)
     (if (<= y 0.08)
       (* (* t_0 (/ (sin x) x)) y)
       (if (<= y 4.2e+116) (sinh y) (* (/ (* t_0 (sin x)) x) y))))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.16666666666666666, 1.0);
	double tmp;
	if (y <= -5800.0) {
		tmp = sinh(y);
	} else if (y <= 0.08) {
		tmp = (t_0 * (sin(x) / x)) * y;
	} else if (y <= 4.2e+116) {
		tmp = sinh(y);
	} else {
		tmp = ((t_0 * sin(x)) / x) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), 0.16666666666666666, 1.0)
	tmp = 0.0
	if (y <= -5800.0)
		tmp = sinh(y);
	elseif (y <= 0.08)
		tmp = Float64(Float64(t_0 * Float64(sin(x) / x)) * y);
	elseif (y <= 4.2e+116)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(Float64(t_0 * sin(x)) / x) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]}, If[LessEqual[y, -5800.0], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 0.08], N[(N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 4.2e+116], N[Sinh[y], $MachinePrecision], N[(N[(N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5800 or 0.0800000000000000017 < y < 4.2000000000000002e116

   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 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 80.7%

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

   9. Applied rewrites 80.7%

      \[\leadsto \sinh y \]

   if -5800 < y < 0.0800000000000000017

   1. Initial program 77.3%

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

   4. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \sinh \left(3 \cdot y\right)\right) \cdot \sin x}{\left(2 \cdot x\right) \cdot \mathsf{fma}\left(2, \cosh \left(2 \cdot y\right), 1\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   7. Applied rewrites 99.0%

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

   9. Applied rewrites 99.0%

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

   if 4.2000000000000002e116 < y

   1. Initial program 100.0%

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

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \sinh \left(3 \cdot y\right)\right) \cdot \sin x}{\left(2 \cdot x\right) \cdot \mathsf{fma}\left(2, \cosh \left(2 \cdot y\right), 1\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   7. Applied rewrites 92.7%

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

Alternative 5: 78.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-12}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.55e-12)
   (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 <= 1.55e-12) {
		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 <= 1.55e-12)
		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, 1.55e-12], 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 < 1.5500000000000001e-12

   1. Initial program 83.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 48.3

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

   5. Applied rewrites 48.3%

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

   7. Applied rewrites 75.9%

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

   9. Applied rewrites 75.9%

      \[\leadsto \sinh y \]

   if 1.5500000000000001e-12 < 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 85.9%

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

Alternative 6: 78.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)}{x} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2e-16)
   (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 <= 2e-16) {
		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 <= 2e-16)
		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, 2e-16], 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 < 2e-16

   1. Initial program 83.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 48.3

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

   5. Applied rewrites 48.3%

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

   7. Applied rewrites 75.9%

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

   9. Applied rewrites 75.9%

      \[\leadsto \sinh y \]

   if 2e-16 < 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 81.4%

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

Alternative 7: 78.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-12}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.55e-12)
   (sinh y)
   (*
    (* (sin x) y)
    (/
     (fma (fma 0.008333333333333333 (* y y) 0.16666666666666666) (* y y) 1.0)
     x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.55e-12) {
		tmp = sinh(y);
	} else {
		tmp = (sin(x) * y) * (fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) / x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.55e-12)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(sin(x) * y) * Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.55e-12], N[Sinh[y], $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * y), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5500000000000001e-12

   1. Initial program 83.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 48.3

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

   5. Applied rewrites 48.3%

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

   7. Applied rewrites 75.9%

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

   9. Applied rewrites 75.9%

      \[\leadsto \sinh y \]

   if 1.5500000000000001e-12 < 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 81.4%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)}{x} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 81.4%

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

Alternative 8: 65.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{+129}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - e^{-y}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.2e+129) (sinh y) (* (- (+ 1.0 y) (exp (- y))) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.2e+129) {
		tmp = sinh(y);
	} else {
		tmp = ((1.0 + y) - exp(-y)) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 4.2d+129) then
        tmp = sinh(y)
    else
        tmp = ((1.0d0 + y) - exp(-y)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4.2e+129) {
		tmp = Math.sinh(y);
	} else {
		tmp = ((1.0 + y) - Math.exp(-y)) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4.2e+129:
		tmp = math.sinh(y)
	else:
		tmp = ((1.0 + y) - math.exp(-y)) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.2e+129)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(Float64(1.0 + y) - exp(Float64(-y))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4.2e+129)
		tmp = sinh(y);
	else
		tmp = ((1.0 + y) - exp(-y)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.2e+129], N[Sinh[y], $MachinePrecision], N[(N[(N[(1.0 + y), $MachinePrecision] - N[Exp[(-y)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.19999999999999993e129

   1. Initial program 85.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 46.6

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

   5. Applied rewrites 46.6%

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

   7. Applied rewrites 71.4%

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

   9. Applied rewrites 71.4%

      \[\leadsto \sinh y \]

   if 4.19999999999999993e129 < 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 59.2

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

   5. Applied rewrites 59.2%

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

   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 56.8%

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

Alternative 9: 65.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+147}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.6e+147)
   (sinh 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 <= 1.6e+147) {
		tmp = sinh(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 <= 1.6e+147)
		tmp = sinh(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, 1.6e+147], N[Sinh[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 < 1.59999999999999989e147

   1. Initial program 85.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 46.5

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

   5. Applied rewrites 46.5%

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

   7. Applied rewrites 70.7%

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

   9. Applied rewrites 70.7%

      \[\leadsto \sinh y \]

   if 1.59999999999999989e147 < 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 61.8

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

   5. Applied rewrites 61.8%

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

   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 50.2%

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

   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 47.4%

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

   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 56.5%

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

Alternative 10: 59.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.18 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 8.7 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right) \cdot x\right) \cdot x - 0.16666666666666666, x \cdot x, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.18e+57)
   (*
    (fma (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y) y 1.0)
    y)
   (if (<= x 8.7e+142)
     (*
      (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 <= 1.18e+57) {
		tmp = fma((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y), y, 1.0) * y;
	} else if (x <= 8.7e+142) {
		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 <= 1.18e+57)
		tmp = Float64(fma(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y), y, 1.0) * y);
	elseif (x <= 8.7e+142)
		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, 1.18e+57], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 8.7e+142], 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 < 1.18e57

   1. Initial program 84.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 89.6%

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

   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 65.0%

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

   10. Applied rewrites 65.0%

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

   if 1.18e57 < x < 8.6999999999999999e142

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   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 22.8

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

   5. Applied rewrites 22.8%

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

   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 53.7%

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

   if 8.6999999999999999e142 < 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 61.4

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

   5. Applied rewrites 61.4%

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

   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 47.7%

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

   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 45.0%

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

   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 53.6%

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

Alternative 11: 61.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+135}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.12e+135)
   (*
    (*
     (fma
      (fma
       (fma 0.0003968253968253968 (* y y) 0.016666666666666666)
       (* y y)
       0.3333333333333333)
      (* y y)
      2.0)
     y)
    0.5)
   (* (- (+ 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 <= 1.12e+135) {
		tmp = (fma(fma(fma(0.0003968253968253968, (y * y), 0.016666666666666666), (y * y), 0.3333333333333333), (y * y), 2.0) * y) * 0.5;
	} 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 <= 1.12e+135)
		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);
	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, 1.12e+135], 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], 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 < 1.1199999999999999e135

   1. Initial program 85.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 46.9

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

   5. Applied rewrites 46.9%

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

   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 65.3%

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

   if 1.1199999999999999e135 < 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 58.1

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

   5. Applied rewrites 58.1%

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

   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 45.1%

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

   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 42.6%

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

   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 50.8%

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

Alternative 12: 59.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.12e+135)
   (*
    (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 <= 1.12e+135) {
		tmp = 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 <= 1.12e+135)
		tmp = Float64(fma(Float64(fma(Float64(y * y), 0.008333333333333333, 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, 1.12e+135], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 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 < 1.1199999999999999e135

   1. Initial program 85.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 88.4%

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

   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 63.1%

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

   10. Applied rewrites 63.1%

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

   if 1.1199999999999999e135 < 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 58.1

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

   5. Applied rewrites 58.1%

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

   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 45.1%

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

   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 42.6%

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

   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 50.8%

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

Alternative 13: 59.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.008333333333333333 \cdot y\right) \cdot y, y \cdot y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.12e+135)
   (* (fma (* (* 0.008333333333333333 y) y) (* 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 <= 1.12e+135) {
		tmp = fma(((0.008333333333333333 * y) * y), (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 <= 1.12e+135)
		tmp = Float64(fma(Float64(Float64(0.008333333333333333 * y) * y), 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, 1.12e+135], N[(N[(N[(N[(0.008333333333333333 * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $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 2 regimes

2. if x < 1.1199999999999999e135

   1. Initial program 85.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 88.4%

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

   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 63.1%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2}, y \cdot y, 1\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 63.0%

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

   13. Applied rewrites 63.0%

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

   if 1.1199999999999999e135 < 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 58.1

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

   5. Applied rewrites 58.1%

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

   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 45.1%

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

   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 42.6%

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

   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 50.8%

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

Alternative 14: 55.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.22e+30)
   (* (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 <= 1.22e+30) {
		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 <= 1.22e+30)
		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, 1.22e+30], 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 < 1.22e30

   1. Initial program 84.1%

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

   4. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \sinh \left(3 \cdot y\right)\right) \cdot \sin x}{\left(2 \cdot x\right) \cdot \mathsf{fma}\left(2, \cosh \left(2 \cdot y\right), 1\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   7. Applied rewrites 82.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.5%

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

   if 1.22e30 < 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 51.4

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

   5. Applied rewrites 51.4%

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

   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 36.3%

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

   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 29.8%

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

   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 41.9%

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

Alternative 15: 56.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.18 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, 1\right), y, 1\right) - \left(1 - y\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.18e+57)
   (* (fma (* y y) 0.16666666666666666 1.0) y)
   (* (- (fma (fma 0.5 y 1.0) y 1.0) (- 1.0 y)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.18e+57) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * y;
	} else {
		tmp = (fma(fma(0.5, y, 1.0), y, 1.0) - (1.0 - y)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.18e+57)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y);
	else
		tmp = Float64(Float64(fma(fma(0.5, y, 1.0), y, 1.0) - Float64(1.0 - y)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.18e+57], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(0.5 * y + 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] - N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.18e57

   1. Initial program 84.6%

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

   4. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \sinh \left(3 \cdot y\right)\right) \cdot \sin x}{\left(2 \cdot x\right) \cdot \mathsf{fma}\left(2, \cosh \left(2 \cdot y\right), 1\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 59.5%

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

   if 1.18e57 < 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 51.5

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

   5. Applied rewrites 51.5%

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

   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 38.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 40.0%

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

Alternative 16: 53.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.12e+135)
   (* (fma (* y y) 0.16666666666666666 1.0) y)
   (* (- (+ 1.0 y) (- 1.0 y)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.12e+135) {
		tmp = fma((y * y), 0.16666666666666666, 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 <= 1.12e+135)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 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, 1.12e+135], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 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 < 1.1199999999999999e135

   1. Initial program 85.5%

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

   4. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \sinh \left(3 \cdot y\right)\right) \cdot \sin x}{\left(2 \cdot x\right) \cdot \mathsf{fma}\left(2, \cosh \left(2 \cdot y\right), 1\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{3}{2} \cdot \frac{\sin x}{x} - \frac{4}{3} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   7. Applied rewrites 80.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 57.4%

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

   if 1.1199999999999999e135 < 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 58.1

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

   5. Applied rewrites 58.1%

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

   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 45.1%

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

   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 42.6%

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

Alternative 17: 36.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.08 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.08e+143)
   (* (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 <= 1.08e+143) {
		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 <= 1.08e+143)
		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, 1.08e+143], 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 < 1.07999999999999993e143

   1. Initial program 85.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 55.2

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

   5. Applied rewrites 55.2%

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

   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 41.7%

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

   if 1.07999999999999993e143 < 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 61.4

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

   5. Applied rewrites 61.4%

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

   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 47.7%

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

   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 45.0%

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

Alternative 18: 32.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+22}:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8e+22) (* 1.0 y) (* (- (+ 1.0 y) (- 1.0 y)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8e+22) {
		tmp = 1.0 * y;
	} else {
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 8d+22) 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 <= 8e+22) {
		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 <= 8e+22:
		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 <= 8e+22)
		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 <= 8e+22)
		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, 8e+22], 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 < 8e22

   1. Initial program 84.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 57.7

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 40.1%

      \[\leadsto 1 \cdot y \]

   if 8e22 < 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 50.6

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

   5. Applied rewrites 50.6%

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

   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 35.8%

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

   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 29.4%

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

Alternative 19: 27.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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 87.6%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 56.1

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

5. Applied rewrites 56.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 31.8%

   \[\leadsto 1 \cdot y \]
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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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 2024363 
(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))