Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.3% → 99.8%

Time: 5.7s

Alternatives: 16

Speedup: 1.4×

• 48.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 88.3% 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 89.6%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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: 68.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-102} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-231}\right):\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(y \cdot y, \left(0.008333333333333333 \cdot y\right) \cdot y, 1\right) \cdot x\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (or (<= t_0 -2e-102) (not (<= t_0 5e-231)))
     (/ (* (* (fma (* y y) (* (* 0.008333333333333333 y) y) 1.0) x) y) x)
     (* (/ y x) x))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if ((t_0 <= -2e-102) || !(t_0 <= 5e-231)) {
		tmp = ((fma((y * y), ((0.008333333333333333 * y) * y), 1.0) * x) * y) / x;
	} else {
		tmp = (y / x) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if ((t_0 <= -2e-102) || !(t_0 <= 5e-231))
		tmp = Float64(Float64(Float64(fma(Float64(y * y), Float64(Float64(0.008333333333333333 * y) * y), 1.0) * x) * y) / x);
	else
		tmp = Float64(Float64(y / x) * x);
	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[Or[LessEqual[t$95$0, -2e-102], N[Not[LessEqual[t$95$0, 5e-231]], $MachinePrecision]], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(0.008333333333333333 * y), $MachinePrecision] * y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999987e-102 or 5.00000000000000023e-231 < (/.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

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 64.4%

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

   if -1.99999999999999987e-102 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5.00000000000000023e-231

   1. Initial program 73.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 14.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 70.2

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

   10. Applied rewrites 70.2%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-102} \lor \neg \left(\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-231}\right):\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(y \cdot y, \left(0.008333333333333333 \cdot y\right) \cdot y, 1\right) \cdot x\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \sinh y}{x}\\ t_1 := \frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \sin x\right) \cdot y}{x}\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -0.012:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-21}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (sinh y)) x))
        (t_1 (/ (* (* (* (* y y) 0.16666666666666666) (sin x)) y) x)))
   (if (<= y -3.6e+109)
     t_1
     (if (<= y -0.012)
       t_0
       (if (<= y 1.15e-21)
         (* (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y) (sin x))
         (if (<= y 1.05e+103) t_0 t_1))))))

C

double code(double x, double y) {
	double t_0 = (x * sinh(y)) / x;
	double t_1 = ((((y * y) * 0.16666666666666666) * sin(x)) * y) / x;
	double tmp;
	if (y <= -3.6e+109) {
		tmp = t_1;
	} else if (y <= -0.012) {
		tmp = t_0;
	} else if (y <= 1.15e-21) {
		tmp = ((fma((y * y), 0.16666666666666666, 1.0) / x) * y) * sin(x);
	} else if (y <= 1.05e+103) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * sinh(y)) / x)
	t_1 = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * sin(x)) * y) / x)
	tmp = 0.0
	if (y <= -3.6e+109)
		tmp = t_1;
	elseif (y <= -0.012)
		tmp = t_0;
	elseif (y <= 1.15e-21)
		tmp = Float64(Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) / x) * y) * sin(x));
	elseif (y <= 1.05e+103)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -3.6e+109], t$95$1, If[LessEqual[y, -0.012], t$95$0, If[LessEqual[y, 1.15e-21], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e+103], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.6e109 or 1.0500000000000001e103 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 100.0%

       \[\leadsto \frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \sin x\right) \cdot y}{x} \]

   if -3.6e109 < y < -0.012 or 1.15e-21 < y < 1.0500000000000001e103

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.8%

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

   if -0.012 < y < 1.15e-21

   1. Initial program 79.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 99.7%

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

Alternative 4: 95.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \sinh y}{x}\\ t_1 := \frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \sin x\right) \cdot y}{x}\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -0.00042:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (sinh y)) x))
        (t_1 (/ (* (* (* (* y y) 0.16666666666666666) (sin x)) y) x)))
   (if (<= y -3.6e+109)
     t_1
     (if (<= y -0.00042)
       t_0
       (if (<= y 2.7e-17)
         (* (/ (sin x) x) y)
         (if (<= y 1.05e+103) t_0 t_1))))))

C

double code(double x, double y) {
	double t_0 = (x * sinh(y)) / x;
	double t_1 = ((((y * y) * 0.16666666666666666) * sin(x)) * y) / x;
	double tmp;
	if (y <= -3.6e+109) {
		tmp = t_1;
	} else if (y <= -0.00042) {
		tmp = t_0;
	} else if (y <= 2.7e-17) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 1.05e+103) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * sinh(y)) / x
    t_1 = ((((y * y) * 0.16666666666666666d0) * sin(x)) * y) / x
    if (y <= (-3.6d+109)) then
        tmp = t_1
    else if (y <= (-0.00042d0)) then
        tmp = t_0
    else if (y <= 2.7d-17) then
        tmp = (sin(x) / x) * y
    else if (y <= 1.05d+103) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * Math.sinh(y)) / x;
	double t_1 = ((((y * y) * 0.16666666666666666) * Math.sin(x)) * y) / x;
	double tmp;
	if (y <= -3.6e+109) {
		tmp = t_1;
	} else if (y <= -0.00042) {
		tmp = t_0;
	} else if (y <= 2.7e-17) {
		tmp = (Math.sin(x) / x) * y;
	} else if (y <= 1.05e+103) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * math.sinh(y)) / x
	t_1 = ((((y * y) * 0.16666666666666666) * math.sin(x)) * y) / x
	tmp = 0
	if y <= -3.6e+109:
		tmp = t_1
	elif y <= -0.00042:
		tmp = t_0
	elif y <= 2.7e-17:
		tmp = (math.sin(x) / x) * y
	elif y <= 1.05e+103:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * sinh(y)) / x)
	t_1 = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * sin(x)) * y) / x)
	tmp = 0.0
	if (y <= -3.6e+109)
		tmp = t_1;
	elseif (y <= -0.00042)
		tmp = t_0;
	elseif (y <= 2.7e-17)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 1.05e+103)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * sinh(y)) / x;
	t_1 = ((((y * y) * 0.16666666666666666) * sin(x)) * y) / x;
	tmp = 0.0;
	if (y <= -3.6e+109)
		tmp = t_1;
	elseif (y <= -0.00042)
		tmp = t_0;
	elseif (y <= 2.7e-17)
		tmp = (sin(x) / x) * y;
	elseif (y <= 1.05e+103)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -3.6e+109], t$95$1, If[LessEqual[y, -0.00042], t$95$0, If[LessEqual[y, 2.7e-17], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.05e+103], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.6e109 or 1.0500000000000001e103 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 100.0%

       \[\leadsto \frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \sin x\right) \cdot y}{x} \]

   if -3.6e109 < y < -4.2000000000000002e-4 or 2.7000000000000001e-17 < y < 1.0500000000000001e103

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.3%

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

   if -4.2000000000000002e-4 < y < 2.7000000000000001e-17

   1. Initial program 79.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.6%

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

Alternative 5: 88.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-23}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7e-23)
   (* (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y) (sin x))
   (/
    (*
     (*
      (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 <= 7e-23) {
		tmp = ((fma((y * y), 0.16666666666666666, 1.0) / x) * y) * sin(x);
	} 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 <= 7e-23)
		tmp = Float64(Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) / x) * y) * sin(x));
	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, 7e-23], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $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 < 6.99999999999999987e-23

   1. Initial program 85.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 88.8%

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

   if 6.99999999999999987e-23 < 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

   5. Applied rewrites 94.4%

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

Alternative 6: 87.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \sinh y}{x}\\ \mathbf{if}\;y \leq -0.00042:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+183}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot y\right) \cdot y\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (sinh y)) x)))
   (if (<= y -0.00042)
     t_0
     (if (<= y 2.7e-17)
       (* (/ (sin x) x) y)
       (if (<= y 2e+183)
         t_0
         (/
          (*
           (*
            (* (* (fma -0.027777777777777776 (* x x) 0.16666666666666666) x) y)
            y)
           y)
          x))))))

C

double code(double x, double y) {
	double t_0 = (x * sinh(y)) / x;
	double tmp;
	if (y <= -0.00042) {
		tmp = t_0;
	} else if (y <= 2.7e-17) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 2e+183) {
		tmp = t_0;
	} else {
		tmp = ((((fma(-0.027777777777777776, (x * x), 0.16666666666666666) * x) * y) * y) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * sinh(y)) / x)
	tmp = 0.0
	if (y <= -0.00042)
		tmp = t_0;
	elseif (y <= 2.7e-17)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 2e+183)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.027777777777777776, Float64(x * x), 0.16666666666666666) * x) * y) * y) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -0.00042], t$95$0, If[LessEqual[y, 2.7e-17], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2e+183], t$95$0, N[(N[(N[(N[(N[(N[(-0.027777777777777776 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.2000000000000002e-4 or 2.7000000000000001e-17 < y < 1.99999999999999989e183

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 78.2%

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

   if -4.2000000000000002e-4 < y < 2.7000000000000001e-17

   1. Initial program 79.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.6%

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

   if 1.99999999999999989e183 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 0.0%

       \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.027777777777777776, y \cdot y, -0.16666666666666666\right) \cdot x, x, \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\right) \cdot x\right) \cdot y}{x} \]

   12. Taylor expanded in y around inf

       \[\leadsto \frac{\left(x \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) \cdot y}{x} \]
   13. Step-by-step derivation

   14. Applied rewrites 82.6%

       \[\leadsto \frac{\left(\left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot y\right) \cdot y\right) \cdot y}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right), 1\right) \cdot x\right) \cdot y}{x}\\ \mathbf{if}\;y \leq -0.00042:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+183}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot y\right) \cdot y\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (*
           (*
            (fma
             (* y y)
             (fma (* 0.008333333333333333 y) y 0.16666666666666666)
             1.0)
            x)
           y)
          x)))
   (if (<= y -0.00042)
     t_0
     (if (<= y 2.7e-17)
       (* (/ (sin x) x) y)
       (if (<= y 2e+183)
         t_0
         (/
          (*
           (*
            (* (* (fma -0.027777777777777776 (* x x) 0.16666666666666666) x) y)
            y)
           y)
          x))))))

C

double code(double x, double y) {
	double t_0 = ((fma((y * y), fma((0.008333333333333333 * y), y, 0.16666666666666666), 1.0) * x) * y) / x;
	double tmp;
	if (y <= -0.00042) {
		tmp = t_0;
	} else if (y <= 2.7e-17) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 2e+183) {
		tmp = t_0;
	} else {
		tmp = ((((fma(-0.027777777777777776, (x * x), 0.16666666666666666) * x) * y) * y) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(fma(Float64(y * y), fma(Float64(0.008333333333333333 * y), y, 0.16666666666666666), 1.0) * x) * y) / x)
	tmp = 0.0
	if (y <= -0.00042)
		tmp = t_0;
	elseif (y <= 2.7e-17)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 2e+183)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.027777777777777776, Float64(x * x), 0.16666666666666666) * x) * y) * y) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -0.00042], t$95$0, If[LessEqual[y, 2.7e-17], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2e+183], t$95$0, N[(N[(N[(N[(N[(N[(-0.027777777777777776 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.2000000000000002e-4 or 2.7000000000000001e-17 < y < 1.99999999999999989e183

   1. Initial program 100.0%

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.5%

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

   if -4.2000000000000002e-4 < y < 2.7000000000000001e-17

   1. Initial program 79.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.6%

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

   if 1.99999999999999989e183 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 0.0%

       \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.027777777777777776, y \cdot y, -0.16666666666666666\right) \cdot x, x, \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\right) \cdot x\right) \cdot y}{x} \]

   12. Taylor expanded in y around inf

       \[\leadsto \frac{\left(x \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) \cdot y}{x} \]
   13. Step-by-step derivation

   14. Applied rewrites 82.6%

       \[\leadsto \frac{\left(\left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot y\right) \cdot y\right) \cdot y}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right), 1\right) \cdot x\right) \cdot y}{x}\\ \mathbf{if}\;y \leq -2 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-53}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+183}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot y\right) \cdot y\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (*
           (*
            (fma
             (* y y)
             (fma (* 0.008333333333333333 y) y 0.16666666666666666)
             1.0)
            x)
           y)
          x)))
   (if (<= y -2e-16)
     t_0
     (if (<= y 3.7e-53)
       (* (/ y x) x)
       (if (<= y 2e+183)
         t_0
         (/
          (*
           (*
            (* (* (fma -0.027777777777777776 (* x x) 0.16666666666666666) x) y)
            y)
           y)
          x))))))

C

double code(double x, double y) {
	double t_0 = ((fma((y * y), fma((0.008333333333333333 * y), y, 0.16666666666666666), 1.0) * x) * y) / x;
	double tmp;
	if (y <= -2e-16) {
		tmp = t_0;
	} else if (y <= 3.7e-53) {
		tmp = (y / x) * x;
	} else if (y <= 2e+183) {
		tmp = t_0;
	} else {
		tmp = ((((fma(-0.027777777777777776, (x * x), 0.16666666666666666) * x) * y) * y) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(fma(Float64(y * y), fma(Float64(0.008333333333333333 * y), y, 0.16666666666666666), 1.0) * x) * y) / x)
	tmp = 0.0
	if (y <= -2e-16)
		tmp = t_0;
	elseif (y <= 3.7e-53)
		tmp = Float64(Float64(y / x) * x);
	elseif (y <= 2e+183)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.027777777777777776, Float64(x * x), 0.16666666666666666) * x) * y) * y) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -2e-16], t$95$0, If[LessEqual[y, 3.7e-53], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 2e+183], t$95$0, N[(N[(N[(N[(N[(N[(-0.027777777777777776 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e-16 or 3.69999999999999982e-53 < y < 1.99999999999999989e183

   1. Initial program 100.0%

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.4%

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

   if -2e-16 < y < 3.69999999999999982e-53

   1. Initial program 77.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 24.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 71.6

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

   10. Applied rewrites 71.6%

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

   if 1.99999999999999989e183 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 0.0%

       \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.027777777777777776, y \cdot y, -0.16666666666666666\right) \cdot x, x, \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\right) \cdot x\right) \cdot y}{x} \]

   12. Taylor expanded in y around inf

       \[\leadsto \frac{\left(x \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) \cdot y}{x} \]
   13. Step-by-step derivation

   14. Applied rewrites 82.6%

       \[\leadsto \frac{\left(\left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot y\right) \cdot y\right) \cdot y}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\mathsf{fma}\left(y \cdot y, \left(0.008333333333333333 \cdot y\right) \cdot y, 1\right) \cdot x\right) \cdot y}{x}\\ \mathbf{if}\;y \leq -2 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-53}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+183}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot y\right) \cdot y\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/ (* (* (fma (* y y) (* (* 0.008333333333333333 y) y) 1.0) x) y) x)))
   (if (<= y -2e-16)
     t_0
     (if (<= y 3.7e-53)
       (* (/ y x) x)
       (if (<= y 2e+183)
         t_0
         (/
          (*
           (*
            (* (* (fma -0.027777777777777776 (* x x) 0.16666666666666666) x) y)
            y)
           y)
          x))))))

C

double code(double x, double y) {
	double t_0 = ((fma((y * y), ((0.008333333333333333 * y) * y), 1.0) * x) * y) / x;
	double tmp;
	if (y <= -2e-16) {
		tmp = t_0;
	} else if (y <= 3.7e-53) {
		tmp = (y / x) * x;
	} else if (y <= 2e+183) {
		tmp = t_0;
	} else {
		tmp = ((((fma(-0.027777777777777776, (x * x), 0.16666666666666666) * x) * y) * y) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(fma(Float64(y * y), Float64(Float64(0.008333333333333333 * y) * y), 1.0) * x) * y) / x)
	tmp = 0.0
	if (y <= -2e-16)
		tmp = t_0;
	elseif (y <= 3.7e-53)
		tmp = Float64(Float64(y / x) * x);
	elseif (y <= 2e+183)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.027777777777777776, Float64(x * x), 0.16666666666666666) * x) * y) * y) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(0.008333333333333333 * y), $MachinePrecision] * y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -2e-16], t$95$0, If[LessEqual[y, 3.7e-53], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 2e+183], t$95$0, N[(N[(N[(N[(N[(N[(-0.027777777777777776 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e-16 or 3.69999999999999982e-53 < y < 1.99999999999999989e183

   1. Initial program 100.0%

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 62.8%

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

   if -2e-16 < y < 3.69999999999999982e-53

   1. Initial program 77.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 24.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 71.6

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

   10. Applied rewrites 71.6%

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

   if 1.99999999999999989e183 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 0.0%

       \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.027777777777777776, y \cdot y, -0.16666666666666666\right) \cdot x, x, \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\right) \cdot x\right) \cdot y}{x} \]

   12. Taylor expanded in y around inf

       \[\leadsto \frac{\left(x \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) \cdot y}{x} \]
   13. Step-by-step derivation

   14. Applied rewrites 82.6%

       \[\leadsto \frac{\left(\left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot y\right) \cdot y\right) \cdot y}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{elif}\;x \leq 4500:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \left(x \cdot y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.2e-81)
   (* (/ y x) x)
   (if (<= x 4500.0)
     (/ (* (fma 0.16666666666666666 (* y y) 1.0) (* x y)) x)
     (/ (* (* (* (* y y) 0.16666666666666666) x) y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.2e-81) {
		tmp = (y / x) * x;
	} else if (x <= 4500.0) {
		tmp = (fma(0.16666666666666666, (y * y), 1.0) * (x * y)) / x;
	} else {
		tmp = ((((y * y) * 0.16666666666666666) * x) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.2e-81)
		tmp = Float64(Float64(y / x) * x);
	elseif (x <= 4500.0)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * Float64(x * y)) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * x) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.2e-81], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 4500.0], N[(N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.1999999999999997e-81

   1. Initial program 85.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 25.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.5

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

   10. Applied rewrites 55.5%

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

   if 7.1999999999999997e-81 < x < 4500

   1. Initial program 92.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 68.3%

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

   13. Applied rewrites 68.3%

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

   if 4500 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 25.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 88.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 22.5%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 44.6%

       \[\leadsto \frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 57.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{elif}\;x \leq 4500:\\ \;\;\;\;\frac{\left(y \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right) \cdot x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.2e-81)
   (* (/ y x) x)
   (if (<= x 4500.0)
     (/ (* (* y (fma 0.16666666666666666 (* y y) 1.0)) x) x)
     (/ (* (* (* (* y y) 0.16666666666666666) x) y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.2e-81) {
		tmp = (y / x) * x;
	} else if (x <= 4500.0) {
		tmp = ((y * fma(0.16666666666666666, (y * y), 1.0)) * x) / x;
	} else {
		tmp = ((((y * y) * 0.16666666666666666) * x) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.2e-81)
		tmp = Float64(Float64(y / x) * x);
	elseif (x <= 4500.0)
		tmp = Float64(Float64(Float64(y * fma(0.16666666666666666, Float64(y * y), 1.0)) * x) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * x) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.2e-81], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 4500.0], N[(N[(N[(y * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.1999999999999997e-81

   1. Initial program 85.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 25.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.5

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

   10. Applied rewrites 55.5%

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

   if 7.1999999999999997e-81 < x < 4500

   1. Initial program 92.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 68.3%

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

   13. Applied rewrites 68.3%

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

   if 4500 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 25.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 88.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 22.5%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 44.6%

       \[\leadsto \frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 52.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-9} \lor \neg \left(x \leq 7.6 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 1e-9) (not (<= x 7.6e+146)))
   (* (/ y x) x)
   (*
    (fma (fma 0.008333333333333333 (* x x) -0.16666666666666666) (* x x) 1.0)
    y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 1e-9) || !(x <= 7.6e+146)) {
		tmp = (y / x) * x;
	} else {
		tmp = fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * x), 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 1e-9) || !(x <= 7.6e+146))
		tmp = Float64(Float64(y / x) * x);
	else
		tmp = Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 1e-9], N[Not[LessEqual[x, 7.6e+146]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000006e-9 or 7.59999999999999958e146 < x

   1. Initial program 88.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 42.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 24.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 53.7

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

   10. Applied rewrites 53.7%

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

   if 1.00000000000000006e-9 < x < 7.59999999999999958e146

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 52.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 25.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-9} \lor \neg \left(x \leq 7.6 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4800000:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4800000.0)
   (* (/ y x) x)
   (/ (* (* (* (* y y) 0.16666666666666666) x) y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4800000.0) {
		tmp = (y / x) * x;
	} else {
		tmp = ((((y * y) * 0.16666666666666666) * x) * y) / x;
	}
	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 <= 4800000.0d0) then
        tmp = (y / x) * x
    else
        tmp = ((((y * y) * 0.16666666666666666d0) * x) * y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4800000.0) {
		tmp = (y / x) * x;
	} else {
		tmp = ((((y * y) * 0.16666666666666666) * x) * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4800000.0:
		tmp = (y / x) * x
	else:
		tmp = ((((y * y) * 0.16666666666666666) * x) * y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4800000.0)
		tmp = Float64(Float64(y / x) * x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * x) * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4800000.0)
		tmp = (y / x) * x;
	else
		tmp = ((((y * y) * 0.16666666666666666) * x) * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4800000.0], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.8e6

   1. Initial program 86.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 40.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 26.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.6

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

   10. Applied rewrites 55.6%

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

   if 4.8e6 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 25.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 88.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 22.5%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 44.6%

       \[\leadsto \frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.0% accurate, 12.8× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (y / x) * 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 = (y / x) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 43.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 23.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 48.0

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

10. Applied rewrites 48.0%

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

Alternative 15: 36.5% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (fma -0.16666666666666666 (* x x) 1.0) y))

C

double code(double x, double y) {
	return fma(-0.16666666666666666, (x * x), 1.0) * y;
}

Julia

function code(x, y)
	return Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * y)
end

Wolfram

code[x_, y_] := N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 89.6%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 54.1%

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

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 34.6%

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

Alternative 16: 29.1% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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 = y
end function

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 89.6%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 54.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 27.0%

   \[\leadsto y \]
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 2025018 
(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))